John, Fritz - Partial Differential Equations (1978, 3rd edition, Springer)

Applied Mathematical Sciences I Volume 1
Fritz John
Partial
Differential Equations
Third Edition
Springer-Verlag
New York· Heidelberg· Berlin
Fritz John
Courant Institute of
Mathematical Sciences
New York University
New York, NY 10012
USA
Editors
Fritz John
Lawrence Sirovich
Courant Institute of
Mathematical Sciences
New York University
New York, NY 10012
USA
Division of
Applied Mathematics
Brown University
Providence, RI 02912
USA
Joseph P. LaSalle
Gerald B. Whitham
Division of
Applied Mathematics
Brown University
Providence, RI 02912
USA
Applied Mathematics
Firestone Laboratory
California Institute of Technology
Pasadena, CA 91125
USA
AMS Subject Classifications: 35-02, 35AIO, 35EXX, 35L05, 35LlO
Library of Congress Cataloging in Publication Data
John, Fritz 1910Partial differential equations.
(Applied mathematical sciences; v. I)
Bibliography: p.
Includes index.
I. Differential equations, Partial. I. Title. II. Series.
QAl.A647 vol. I 1978 [QA374] 510'.8s [515'.353]
78-10449
All rights reserved.
No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag.
Copyright © 1971, 1975, 1978 by Springer-Verlag New York Inc.
Softcover reprint ofthe hardcover 3rd edition 1978
ISBN -13: 978-1-4684-0061-8
DOl: 10.1007/978-1-4684-0059-5
e-ISBN -13: 978-1-4684-0059-5
Preface to the Third Edition
The book has been completely rewritten for this new edition. While most
of the material found in the earlier editions has been retained, though in
changed form, there are considerable additions, in which extensive use is
made of Fourier transform techniques, Hilbert space, and finite difference
methods.
A condensed version of the present work was presented in a series of
lectures as part of the Tata Institute of Fundamental Research -Indian Institute of Science Mathematics Programme in Bangalore in 1977. I am indebted
to Professor K. G. Ramanathan for the opportunity to participate in this exciting educational venture, and to Professor K. Balagangadharan for his ever
ready help and advice and many stimulating discussions. Very special thanks
are due to N. Sivaramakrishnan and R. Mythili, who ably and cheerfully
prepared notes of my lectures which I was able to use as the nucleus of the
present edition.
A word about the choice of material. The constraints imposed by a
partial differential equation on its solutions (like those imposed by the
environment on a living organism) have an infinite variety of consequences, local and global, identities and inequalities. Theories of such
equations usually attempt to analyse the structure of individual solutions
and of the whole manifold of solutions by testing the compatibility of the
differential equation with various types of additional constraints. The
problems arising in this way have challenged the ingenuity of mathematicians for centuries. It is good to keep in mind that there is no single
"central" problem; new applications ·commonly lead to new questions
never envisioned before. In this book emphasis is put on discovering
significant features of a differential equation, and on exploring them as far
as possible with a limited amount of machinery from mathematical analysis. Entanglement in a mass of technical details has been avoided, even
when this resulted in less general or less complete results.
New Rochelle, N. Y.
Fritz John
v
Contents
Chapter 1
The single first-order equation
1
I. Introduction
2. Examples
3. Analytic Solution and Approximation Methods in a Simple
Example
Problems
4. Quasi-linear Equations
5. The Cauchy Problem for the Quasi-linear Equation
6. Examples
Problems
7. The General First-order Equation for a Function of Two
Variables
8. The Cauchy Problem
9. Solutions Generated as Envelopes
Problems
I
2
4
8
8
10
14
18
19
23
28
30
Chapter 2
Second-order equations: hyperbolic equations
for functions of two independent variables
l. Characteristics for Linear and Quasi-linear Second-order
Equations
2. Propagation of Singularities
3. The Linear Second-order Equation
Problems
4. The One-Dimensional Wave Equation
Problems
31
31
33
35
37
38
43
vii
Contents
5.
6.
Systems of First-order Equations
(Courant-Lax Theory)
A Quasi-linear System and Simple Waves
Problem
44
50
51
Chapter 3
Characteristic manifolds and the Cauchy problem
52
1. Notation of Laurent Schwartz
Problems
2. The Cauchy Problem
Problems
3. Cauchy-Kowalewski Theorem
Problems
4. The Lagrange-Green Identity
5. The Uniqueness Theorem of Holmgren
6. Distribution Solutions
Problems
52
53
54
59
59
63
64
65
67
70
Chapter 4
The Laplace equation
1. Green's Identity, Fundamental Solutions
Problems
2. The Maximum Principle
Problems
3. The Dirichlet Problem, Green's Function, and Poisson's Formula
Problems
4. Proof of Existence of Solutions for the Dirichlet Problem Using
Subharmonic Functions ("Perron's Method")
Problem
5. Solution of the Dirichlet Problem by Hilbert-Space Methods
Problems
72
72
79
81
83
84
88
89
94
94
101
Chapter 5
Hyperbolic equations in higher dimensions
1.
V1ll
The Wave Equation in n-dimensional Space
(a) The method of spherical means
Problems
(b) Hadamard's method of descent
Problems
(c) Duhamel's principle and the general Cauchy problem
Problem
(d) Initial-boundary-value problems ("Mixed" problems)
Problems
I
103
103
103
109
110
III
112
116
116
119
Contents
2. Higher-order Hyperbolic Equations with Constant Coefficients
(a) Standard form of the initial-value problem
Problem
(b) Solution by Fourier transformation
Problems
(c) Solution of a mixed problem by Fourier transformation
(d) The method of plane waves
Problems
3. Symmetric Hyperbolic Systems
(a) The basic energy inequality
Problems
(b) Existence of solutions by the method of finite differences
Problems
120
120
122
122
132
133
135
138
139
139
145
146
155
Chapter 6
Higher-order elliptic equations with constant
coefficients
1. The Fundamental Solution for Odd n
Problems
2. The Dirichlet Problem
Problems
156
157
159
160
164
Chapter 7
Parabolic equations
1. The Heat Equation
(a) The initial-value problem
Problems
(b) Maximum principle, uniqueness, and regularity
Problems
(c) A mixed problem
Problems
2. The Initial-value Problem for General Second-order Linear
Parabolic Equations
(a) The method of finite differences and the maximum principle
(b) Existence of solutions of the initial-value problem
Problems
166
166
166
173
174
179
179
181
181
181
185
188
Bibliography
191
Glossary
193
Index
195
ix
The single first-order equation *
1
1. Introduction
A partial differential equation (henceforth abbreviated as P.D.E.) for a
function u(x,y, ... ) is a relation of the form
F(x,y, ... ,u,ux,Uy, ... ,uxx,uxy, ... ) =0,
(1.1)
where F is a given function of the independent variables x,y, ... , and of
the "unknown" function u and of a finite number of its partial derivatives.
We call u a solution of (1.1) if after substitution of u(x,y, ... ) and its partial
derivatives (1.1) is satisfied identically in x,y, ... in some region n in the
space of these independent variables. Unless the contrary is stated we
require that x,y, ... are real and that u and the derivatives of u occurring in
(1.1) are continuous functions of x,y, ... in the real domain n.t Several
P.D.E.s involving one or more unknown functions and their derivatives
constitute a system.
The order of a P.D.E. or of a system is the order of the highest
derivative that occurs. A P.D.E. is said to be linear if it is linear in the
unknown functions and their derivatives, with coefficients depending on
the independent variables x,y, .... The P.D.E. of order m is called quasilinear if it is linear in the derivatives of order m with coefficients that
depend on x,y, ... and the derivatives of order < m.
• ([71, [131, [26D
t For simplicity we shall often dispense with an explicit description of the domain
n.
Statements made then apply "locally," in a suitably restricted neighborhood of a point of
X)' ••• -space.
I The single first-order equation
2. Examples
Partial differential equations occur throughout mathematics. In this section
we give some examples. In many instances one of the independent variables is the time, usually denoted by t, while the others, denoted by
X\,X2, ... ,Xn (or by x,y,z when n(;3) give position in an n-dimensional
space. The space differentiations often occur in the particular combination
a=£+ ... +£
axi
ax;
(2.1)
known as the Laplace operator. This operator has the special property of
being invariant under rigid motions or equivalently of not being affected
by transitions to other cartesian coordinate systems. It occurs naturally in
expressing physical laws that do not depend on a special position.
(i) The Laplace equation in n dimensions for a function u(x\, ... ,xn) is
the linear second-order equation
(2.2)
This is probably the most important individual P.D.E. with the widest
range of applications. Solutions u are called potential functions or harmonic
functions. For n=2, x\ =x, X2=Y' we can associate with a harmonic
function u(x,y) a "conjugate" harmonic function v(x,y) such that the
first-order system of Cauchy-Riemann equations
ux = vy,
Uy = - vx
(2.3)
is satisfied. A real solution (u,v) of (2.3) gives rise to the analytic function
fez) = f(x+ ry) =u(x,y)+ iv(x,y)
(2.4)
of the complex argument z = x + ry. We can also interpret (u( x,y),
- v(x,y» as the velocity field of an irrotational, incompressible flow. For
n = 3 equation (2.2) is satisfied by the velocity potential of an irrotational
incompressible flow, by gravitational and electrostatic fields (outside the
attracting masses or charges), and by temperatures in thermal equilibrium.
(ii) The wave equation in n dimensions for u=u(x\, ... ,xn,t) is
(2.5)
(c = const. > 0). It represents vibrations of strings or propagation of sound
waves in tubes for n= 1, waves on the surface of shallow water for n=2,
acoustic or light waves for n = 3.
(iii) Maxwell's equation in vacuum for the electric vector E=(E\,E2,E3)
and magnetic vector H=(H\,H2 ,H3) form a linear system of essentially 6
first-order equations
eEt=curl H,
pHt= -curl E
(2.6a)
divE=divH=O
2
(2.6b)
2 Examples
with constants e, p.. (If relations (2.6b) hold for t = 0, they hold for all t as a
consequence of relations (2.6a». Here each component E;,Hk satisfies the
wave equation (2.5) with c 2 = 1/ ep..
(iv) Elastic waves are described classically by the linear system
a
a2u.
P-i- = p.Llui +(A+ p.)-a (divu)
Xi
at
(2.7)
(i= 1,2,3), where the Ui (X 1,X2,x3,t) are the components of the displacement
vector u, and p is the density and A, p. the Lame constants of the elastic
material. Each Ui satisfies the fourth-order equation
_A+2p. Ll)(l.:..
- .l!:Ll)U.=O,
(l.:..
at2
at 2
p
p
(2.8)
I
formed from two different wave operators. For elastic equilibrium (ut=O)
we obtain the biharmonic equation
Ll2u = O.
(2.9)
(v) The equation of heat conduction ("heat equation")
ut=kLlu
(2.10)
(k = const. > 0) is satisfied by the temperature of a body conducting heat,
when the density and specific heat are constant.
(vi) Schrodinger's wave equation (n=3) for a single particle of mass m
moving in a field of potential energy V(x,y,z) is
ilitf;t= - ;:Lltf;+ Vtf;,
(2.11)
where h = 2'1T1i is Planck's constant.
The equations in the preceding examples were all linear. Nonlinear
equations occur just as frequently, but are inherently more difficult, hence
in practice they are often approximated by linear ones. Some examples of
nonlinear equations follow.
(vii) A minimal surface z = u(x,y) (i.e., a surface having least area for a
given contour) satisfies the second-order quasi-linear equation
(1 + uj)uxx -2ux uy uxy +(1 + u;)uyy =0.
(2.12)
(viii) The velocity potential q,(x,y) (for velocity components q,x,q,y) of a
two-dimensional steady, adiabatic, irrotational, isentropic flow of density p
satisfies
(1- c- 2q,;)q,xx - 2c- 2q,Ayq,xy + (1- c- 2q,;)q,yy =0,
(2.13)
where c is a known function of the speed q=yq,;+q,; . For example
y-1
c 2= 1- __ q2
2
for a polytropic gas with equation of state
p=ApY.
(2.14)
(2.15)
3
I The single first-order equation
(ix) The Navier-Stokes equations for the viscous flow of an incompressible liquid connect the velocity components Uk and the pressure p:
au;
-+
at
L -au;u =---+y!:J.u.,
1 ap
k
axk
L
k
k
aUk
p
ax.I
I
=0,
aXk
(2.l6a)
(2. 16b)
where p is the constant density and y the kinematic viscosity.
(x) An example of Ii third-order nonlinear equation for a function u(x,t)
is furnished by the Korteweg-de Vries equation
(2.17)
first encountered in the study of water waves.
In general we shall try to describe the manifold of solutions of a P.D.E.
The results differ widely for different classes of equations. Meaningful
"well-posed" problems associated with a P.D.E. often are suggested by
particular physical interpretations and applications.
3. Analytic Solution and Approximation Methods
in a Simple Example*
We illustrate some of the notions that will play an important role in what
follows by considering one of the simplest of all equations
ut+cux=O
(3.1)
for a function u= u(x,t), where c=const. >0. Along a line of the family
(3.2)
x- ct=const.=~
("characteristic line" in the xt-plane) we have for a solution u of (3.1)
du
d
dt = dtu(ct+~,t)=cux+Ut=O.
Hence u is constant along such a line, and depends only on the parameter
which distinguishes different lines. The general solution of (3.1) then has
the form
(3.3)
u(x, t) =f(~) = f(x - ct).
~
Formula (3.3) represents the general solution u uniquely in terms of its
initial values
(3.4)
u(x,O)= f(x).
Conversely every u of the form (3.3) is a solution of (3.1) with initial values
f providedf is of class C1(R). We notice that the value of u at any point
(x,t) depends only on the initial valuefat the single argument ~=x-ct,
the abscissa of the point of intersection of the characteristic line through
*({16], [18], [2SD
4
3 Analytic Solution and Approximation Methods in a Simple Example
-+-,~----------x
Figure 1.1
(x, t) with the initial line, the x-axis. The domain of dependence of u(x, t) on
the initial values is represented by the single point ~. The influence of the
initial values at a particular point ~ on the solution u(x,t) is felt just in the
points of the characteristic line (3.2). (Fig. 1.1)
If for each fixed t the function u is represented by its graph in the
ut-plane, we find that the graph at the time t = T is obtained by translating
the graph at the time t = 0 parallel to the x-axis by the amount cT:
u(x, 0) = u(x+ cT, T)= f(x).
The graph of the solution represents a wave propagating to the right with
velocity c without changing shape. (Fig. 1.2)
We use this example with its explicit solution to bring out some of the
notions connected with the numerical solution of a P.D.E by the method of
finite differences. One covers the xt-plane by a rectangular grid with mesh
size h in the x-direction and k in the t-direction. In other words one
considers only points (x,t) for which x is a multiple of hand t a multiple
of k. It would seem natural for purposes of numerical approximation to
replace the P.D.E. (3.1) by the difference equation
v(x,t+k)-v(x,t)
v(x+h,t)-v(x,t)
k
+c
h
=0.
(3.5)
Formally this equation goes over into vt+cvx=O as h,k-+O. We ask to
--~------~----------L----x
x
x
+ cT
Figure 1.2
5
1 The single first-order equation
what extent a solution v of (3.5) in the grid points with initial values
v(x,O)= f(x)
(3.6)
approximates for small h,k the solution of the initial-value problem (3.1),
(3.4).
Setting A= k / h, we write (3.5) as a recursion formula
v(x,t+ k)=(1 +AC)V(X,t) -ACV(X+ h,t)
(3.7)
expressing v at the time t+ k in terms of v at the time t. Introducing the
shift operator E defined by
(3.8)
Ef(x) = f(x+ h),
(3.7) becomes
v(x,t+ k)=(1 +AC) -AcE)v(x,t)
(3.8a)
for t = nk this immediately leads by iteration to the solution of the
initial-value problem for (3.5):
v(x,t) = v(x,nk) =(1 +AC) -AcE Yv(x, 0)
n
=
L (::Z)(1 +AC)m( -AcEy-mf(x)
m=O
L (::Z)(1 +AC)m( -Acr-mf(x+(n- m)h).
n
=
(3.9)
m=O
Clearly the domain of dependence for v(x,t)=v(x,nk) consists of the set
of points
x, x+h, x+2h, ... , x+nh
(3.10)
on the x-axis, all of which lie between x and x + nh. The domain of the
differential equation solution consists of the point ~ = x- ct = x - Mh,
which lies completely outside the interval (x,x + nh). It is clear that v for
h,k~O cannot be expected to converge to the correct solution u of the
differential equation, since in forming v(x,t) we do not make use of any
information on the value of f(~, which is vital for determining u(x, t), but
only of more and more information onfin the interval (x,X+(t/A)) which
is irrelevant. The difference scheme fails the Courant-Friedrichs-Lewy test,
which requires that the limit of the domain of dependence for th~ difference equation contains the domain of dependence for the differential
equation.
That the scheme (3.5) is inappropriate also is indicated by its high
degree of instability. In applied problems the data f are never known with
perfect accuracy. Moreover, in numerical computations we cannot easily
use the exact values but commit small round-off errors at every step. Now
it is clear from (3.9) that errors in f of absolute value e with the proper
(alternating) sign can lead to a resulting error in v(x,t)=v(x,nk) of size
n
e
L (::Z)(1 +Ac)m(Acr-m=(1 +2ACre.
m=O
6
(3.11 )
3 Analytic Solution and Approximation Methods in a Simple Example
Thus for a fixed mesh ratio A the possible resulting error in v grows
exponentially with the number n of steps in the t-direction.
A more appropriate difference scheme uses "backward" difference
quotients:
v(x,t+k)-v(x,t)
v(x,t)-v(x-h,t)
k
+c
h
=0
(3.12)
or symbolically
vex, t+ k) = (I-AC) + AcE -1)V(X, t).
(3.13)
The solution of the initial-value problem for (3.13) becomes
n
v(x,t)=v(x,nk)=
:L (::Z)(1-AC)m(Act-mf(x-(n-m)h).
(3.14)
m=O
In this scheme the domain of dependence for
points
vex, t) on f consists of the
x, x-h, x-2h, ... , x-nh=x-
t
X
(3.15)
Letting h, k~O in such a way that the mesh ratio A is held fixed, the set
(3.15) has as its limit points the interval [x-(t/A),x] on the x-axis. The
Courant-Friedrichs-Lewy test is satisfied, when this interval contains the
point ~ = x - ct, that is when the mesh ratio A satisfies
Ac": 1.
(3.16)
Stability of the scheme under the condition (3.16) is indicated by the fact
that by (3.14) a maximum error of size e in the initial functionfresults in a
maximum possible error in the value of v(x,t)=v(x,nk) of size
n
e
:L (::Z)(1-AC)m(Act-m=e(I-AC)+Act=e.
(3.17)
m=O
We can prove that the v represented by (3.14) actually converges to
u(x,t)=f(x-ct) for h,k~O with k/h=A fixed, provided the stability
criterion (3.16) holds andfhas uniformly bounded second derivatives. For
that purpose we observe that u(x, t) satisfies
lu(x,t+ k) -(I-Ac)u(x,t) -Acu(x- h,t)1
= If(x - ct- ck)- (I-Ac)f(x- ct)-Acf(x- ct- h)l..: Kh 2,
(3.18)
where
(3.19)
as is seen by expanding f about the point x - ct. Thus, setting w = u - v we
have
Iw(x,t+ k)-(1-Ac)w(x,t) -Acw(x- h,t)l..: Kh 2
7
1 The single first-order equation
and hence
sup Iw(x,t+ k)1 (;(I-Ac) sup Iw(x,t)1 +ACSUP IW(X- h,t)1 + Kh 2
x
x
x
(3.20)
= sup Iw(x,t)1 +Kh 2•
x
Applying (3.20) repeatedly it follows for t = nk that
lu(x,t) - v(x,t)1 (; sup Iw(x,nk)1
x
2
Kth
(;s~plw(x,O)I+nKh = T '
since w(x,O)=O. Consequently w(x,t)~O as h~O, that is, the solution v of
the difference scheme (3.12) converges to the solution u of the differential
equation.
PROBLEMS
1. Show that the solution v of (3.12) with initial dataJ converges to u for h~O and
a fixed A<; 1/ c, under the sole assumption that J is continuous. (Hint: the fact
that both u and v change by at most e when we change J by at most e.)
2. To take into account possible round-off errors we assume that instead of (3.13) v
satisfies an inequality
Iv(x,t+ k)-(I-Ac)v(x,t)-Acv(x- h,t)1 <8.
Show that for a prescribed 8 and for K given by (3.19) we have the estimate
Kth
t
lu(x,t)- v(x,t)1 <; T + Ah 8
(3.21)
assuming that (3.16) holds and that v(x, 0) =J(x). Find values for A and h based
on this formula that will guarantee the smallest maximum error in computing
u(x,t).
3. Instability of a difference scheme under small perturbations does not exclude
the possibility that in special cases the scheme converges towards the correct
function, if no errors are permitted in the data or the computation. In particular
let J(x) = eax with a complex constant a. Show that for fixed x, t and any fixed
positive A=k/h whatsoever both the expressions (3.9) and (3.14) converge for
n ~ 00 towards the correct limit e a(x - CI). (This is consistent with the
Courant-Friedrichs-Lewy test, since for an analytic J the values of J in any
interval determine those at the point ~ uniquely.)
4. Quasi-linear Equations
The general first-order equation for a function u=u(x,y, ... ,z) has the
form
f(x,y, ... ,u,ux'~'·· .,uz ) =0.
(4.1)
Equations of this type occur naturally in the calculus of variations, in
particle mechanics, and in geometrical optics. The main result is the fact
8
4 Quasi-linear Equations
that the general solution of an equation of type (4.1) can be obtained by
solving systems of Ordinary Differential Equations (O.D.E.s for short).
This is not true for higher-order equations or for systems of first-order
equations. In what follows we shall mostly limit ourselves to the case of
two independent variables x,y. The theory can be extended to more
independent variables without any essential change.
We first consider t~e somewhat simpler case of a quasi-linear equation
a(x,y,u)ux + b(x,y,u)Uy = c(x,y,u).
(4.2)
We represent the function u(x,y) by a surface z = u(x,y) in xyz-space.
Surfaces corresponding to solutions of a P.D.E. are called integral surfaces
of the P.D.E. The prescribed functions a(x,y,z),b(x,y,z),c(x,y,z) define a
field of vectors in xyz-space (or in a portion n of that space). Obviously
only the direction of the vector, the characteristic direction, matters for the
P.D.E. (4.2). Since (ux,Uy, -1) constitute direction numbers of the normal
of the surface z = u(x,y), we see that (4.2) is just the condition that the
normal of an integral surface at any point is perpendicular to the direction
of the vector (a,b,c) corresponding to that point. Thus integral surfaces are
surfaces that at each point are tangent to the characteristic direction.
With the field of characteristic directions with direction numbers (a,b,c)
we associate the family of characteristic curves which at each point are
tangent to that direction field. Along a characteristic curve the relation
~
a(x,y,z)
=
•
b(x,y,z)
=
~
c(x,y,z)
~~
holds. Referring the curve to a suitable parameter t (or denoting the
common ratio in (4.3) by dt) we can write the condition defining characteristic curves in the more familiar form of a system of ordinary differential
equations
':;; = a(x,y,z),
•
dt =b(x,y,z),
:
=c(x,y,z).
(4.4)
The system is "autonomous" (the independent variable t does not appear
explicitly). The choice of the parameter t in (4.4) is artificial. Using any
other parameter along the curve amounts to replacing a,b,c by proportional quantities, which does not change the characteristic curve in xyzspace or the P.D.E. (4.2). Assuming that a,b,c are of class C' in a region
n, we know from the theory of O.D.E.s that through each point of n there
passes exactly one characteristic curve. There is a 2-parameter family of
characteristic curves in xyz-space (but a 3-parameter family of solutions
(x(t),y(t),z(t» of (4.4), since replacing the independent variable t by t+c
with a constant c changes the solution (x,y,z), but not the characteristic
curve, which is its range).
If a surface S: z = u(x,y) is a union of characteristic curves, then S IS an
integral surface. For then through any point P of S there passes a
characteristic curve 'Y contained in S. The tangent to 'Y at P necessarily lies
in the tangent place of S at P. Since the tangent to 'Y has the characteristic
9
1 The single first-order equation
direction, the normal to S at P is perpendicular to the characteristic
direction, which makes S an integral surface. Conversely we can show that
every integral surface S is the union of characteristic curves, or that
through every point of S there passes a characteristic curve contained in S.
This is a consequence of the following theorem:
Theorem. Let the point P = (xo,Yo, zo) lie on the integral surface z = u( x,y).
Let y be the characteristic curve through P. Then y lies completely on S.
PROOF. Let y given by (x(t),y(t),z(t» be the solution of (4.4) for which
(x,y,z) = (xo,yo,zo) for t=to. From y and S we form the expression
U=z(t)-u(x(t),y(t))= U(t).
(4.5)
Obviously U(to) =0 since P lies on S. By (4.4)
dU = dz
dt
dt
- uxCx(t),y(t) ) dx
dt -Ily ( x(t),y(t)) dy
dt
= c(x,y,z) - uxCx,y )a(x,y,z) -Ily(x,y,z)b(x,y,z).
This can be written as the ordinary differential equation
~~ =c(x,y, U+ u(x,y))-ux(x,y)a(x,y, U+ u(x,y))
-Ily(x,y)b(x,y, U+ u(x,y))
(4.6)
for U, where for x,y we have to substitute the functions x(t),y(t) from the
description of y. Now U=O is a particular solution of (4.6), since u(x,y)
satisfies (4.2). By the uniqueness theorem for O.D.E.s, this is the only
solution vanishing for t = to. It follows that the function U (t) defined by
(4.5) vanishes identically. But that just means that the whole curve y lies on
S.
D
As a consequence of the theorem two integral surfaces that have a point
P in common intersect along the whole characteristic curve y through P.
Conversely if the integral surfaces S \ and S2 intersect, without touching,
along a curve y, then y is characteristic. For consider the tangent planes
'17\,'172 to S\,S2 at a point P of y. Each of the planes has to contain the
characteristic direction (a,b,c) at P. Since '17\ =1='172 it follows that the
intersection of '17\ and '172 has the direction (a,b,c). Since the tangent T to y
at P also has to belong to both '17\ and '172' it follows that T has the direction
(a,b,c), and hence that y is characteristic.
S. The Cauchy Problem for the Quasi-linear Equation
We now have a simple description for the general solution u of (4.2): The
integral surface z = u(x,y) is the union of characteristic curves. To get a
better insight into the structure of the manifold of solutions it is desirable
to have a definite method of generating solutions in terms of a prescribed
lO
5 The Cauchy Problem for the Quasi-linear Equation
set F of functions, called "data." Ideally we have a mapping F~u of data
F onto solutions u of the P.D.E. The space of solutions is then described
by the usually simpler space of data. A good deal of the theory of P.D.E.s
is concerned with the "problem" of actually finding the u belonging to a
given F. (Here "finding" commonly is equated with "establishing existence.")
A simple way of selecting an individual u(x,y) out of the infinite set of
all solutions of (4.2) consists in prescribing a curve r in xyz-space which is
to be contained in the integral surface z = u(x,y). Let r be represented
parametrically by
(5.1)
x= f(s),
y=g(s),
z=h(s).
We are asking for a solution u(x,y) of (4.2) such that the relation
>
h(s) = u(j(s),g(s»)
(5.2)
holds identically in s. Finding the function u(x,y) for given dataf(s), g(s),
h(s) constitutes the Cauchy problem for (4.2). Actually the same curve r
has many different parametric representations (5.1) for different choices of
the parameter s. Introducing a different parameter (1 by a substitution
s=q,«(1) will not change the solution u(x,y) of the Cauchy problem.
We shall be satisfied here with a local solution u of our problem,
defined for x,y near values Xo = f(so), Yo = g(so).
In many instances the variable y will be identified with time and x with
position in space. It is then natural to pose the problem of finding a
solution u(x,y) from its initial values at the time y =0:
(5.3)
This initial-value problem obviously is the special Cauchy problem in
which the curve r has the form
(5.4)
z = h(s),
y=O,
x=s,
that is, r lies in the xz-plane and is referred to x as parameter. We notice
that in the initial-value problem we prescribe a single function h(x), which
in tum is determined uniquely by u, whereas in the general Cauchy
problem many space curves r are bound to lead to the same u. An integral
surface contains many curves r but only one intersection with the x~­
plane.
Let then the functions f(s), g(s), h(s) describing r be of class C l in a
neighborhood of a value so. Let
u(x,O)=h(x).
Po = (xo,yo,zo) = (j(so),g(so),h(so».
(5.5)
Assume that the coefficients a(x,y,z), b(x,y,z), c(x,y,z) in (4.2) are of
class C I in x,y,z near Po. It is clear intuitively that the integral surface
z = u(x,y) passing through r will have to consist of the characteristic
curves passing through the various points of r. Accordingly we form for
each s near So that solution
x=X(s,t),
y= Y(s,t),
z=Z(s,t)
(5.6)
11
1 The single first-order equation
of the characteristic differential equations (4.4) which reduces to
j(s),g(s),h(s) for t=O. The functions X, Y,Z then satisfy
X,=a(X,Y,Z),
~=b(X,Y,Z),
Z,=c(X,Y,Z)
(5.7)
identically in s, t and also satisfy the initial conditions
X(s,O)=j(s),
Y(s,O)=g(s),
Z(s,O)=h(s).
(5.8)
From the general theorems on existence and on continuous dependence on
parameters of solutions of systems of ordinary differential equations it
follows that there exists a unique set of functions X(s,t), Y(s,t), Z(s,t) of
class C 1 for (s,t) near (so,O) which satisfy (5.7), (5.8).
Equations (5.6) represent a surface~: z=u(x,y) referred to parameters
s, t if we can solve the first two equations for s, t in terms of x,y, say in the
forms=S(x,y), t=T(x,y). Then the u defined by
7.
= u(x,y) = Z(S (x,y), T(x,y»
will be the explicit representation of
xo=X(so,O),
~.
(5.9)
By (5.5), (5.8)
Yo= Y(so,O).
(5.10)
Now the implicit function theorem asserts that we can find solutions
s=S(x,y), t= T(x,y) of
x=X(S (x,y), T(x,y»,
of class
C1
y = Y(S(x,y), T(x,y»
(5.11)
in a neighborhood of (xo,Yo) and satisfying
So= S (xo,Yo),
0= T(xQoYo),
provided the Jacobian
Ys(so,O)
Yt(so,O)
I
(5.12)
(5.13)
does not vanish. By (5.7), (5.8) this amounts to the condition
J=
I
j'(so)
a(xo,yo,zo)
g'(so)
b(xo,yo,zo)
I~O.
(5.14)
Thus (5.13) guarantees that locally (5.6) represents a surface ~: z = u(x,y).
That ~ is an integral surface is clear in the parametric representation (5.6).
For at any point P the quantities X" Y" Zt give the direction of the tangent
to a curve s = const. on ~, which will have to lie in the tangent plane of ~
at P. Thus (5.7) shows that the tangent plane at any point contains the
characteristic direction (a,b,c), and hence that ~ is an integral surface.
[One can also verify analytically that the function u represented by (5.9)
satisfies the differential equation (4.2) by first expressing ux,lly in terms of
Sx' Sy, Tx, r;" and then expressing those four quantities in terms of
Xs'X" Ys' Y, using (5.11).]
This completes the local existence proof for the solution of the Cauchy
problem, under the assumption (5.13). Uniqueness follows from the theorem on p. 10: Any integral surface through r would have to contain the
12
5 The Cauchy Problem for the Quasi-linear Equation
characteristic curves through the points of f, hence would have to contain
the surface represented parametrically by (5.6), and hence locally would
have to be identical with the surface.
Condition (5.14) is essential for the existence of a Cl-solution u(x,y) of
the Cauchy problem. For if J=O we would find from (5.2), (4.2) that at
s = so' x = f(so), y = g(so) the three relations
bf'-ag'=O,
(5.15)
h'= f'ux + g'~,
hold. These imply that
bh' - cg' = 0,
ah' - cf' =0
and hence thatf',g',h' are proportional to a,b,c. Hence J=O is incompatible with the existence of a solution unless f happens to be characteristic at
so. Incidentally the Cauchy problem will have infinitely many solutions for
a characteristic curve f, which are obtained by passing any curve f*
satisfying (5.14) through a point Po of f and solving the Cauchy problem
for f*.
In the special case of a linear P.D.E. we can write (4.2) in the form
a (x,y)ux + b(x,y)~ = c(x,y)u + d(x,y).
(5.16)
Here the system of three characteristic O.D.E.s reduces to the pair
dx
dy
(5.17)
dt =a(x,y),
dt =b(x,y)
or even to the single equivalent equation
dy
b(x,y)
dx = a(x,y) .
(5.18)
Equations (5.11) or (5.18) determine a system of curves in the xy-plane,
called characteristic projections, (also, more commonly and confusingly,
just "characteristics") which are the projections onto the xy-plane of the
characteristic curves in xyz-space. The characteristic curve is obtained
from its projection x(t),y(t) by finding z(t) from the linear O.D.E.
: = c{x(t),y(t»z + d(x(t),y(t».
(5.19)
We indicate how to proceed in the more general case of a quasi-linear
equation for a function u = u(x l , ... ,xn) of n independent variables. Such
an equation has the form
n
L ai(xI,···,Xn,u)U",,=C(XI,··.,Xn,u).
(5.20)
i= I
Here the characteristic curves in XI ••• xnz-space are given by the system of
O.D.E.s
dx.
(5.2Ia)
d: =ai(xI'···'xn,z) fori=I, ... ,n
: =c(xl,···,xn,z).
(5.2Ib)
13
1 The single first-order equation
In the Cauchy problem we want to pass an integral surface z=u(Xt, ... ,xn)
in ~.n+ t through an (n - I)-dimensional manifold r given parametrically by
Xj
for i= l, ... ,n
= ./;{St, ... ,sn-t)
(5.22a)
z=h{st"",sn_t).
(5.22b)
For that purpose we pass through each point of r with parameters
St"",sn-t a characteristic curve (solution of (5.2Ia, b) reducing to
(jt, ... ,jn,h) for t=O) rep~esented by
for i= l, ... ,n
xj=Xj{St, ... ,sn_t,t)
(5.23a)
z=Z{st>""sn_t,t).
(5.23b)
These equations form a parametric representation for the desired integral
surface z = u(x t, ... , xn), provided relations (5.23a) can be solved for
St, ... ,Sn-t, t. This is the case when the Jacobian
ait
aS t
ain
aS t
ait
aSn_ t
at
ain
aSn- t
an
(5.24)
J=
does not vanish.
6. Examples
(1) (See Section 2.)
Uy
+ cux=O
(6.1a)
(c=const.)
u{x,O)= hex).
The initial curve
r corresponding to (6.lb) is given by
x=s,
y=O,
(6.1b)
z=h{s).
The characteristic differential equations are
dx
-=c
dt
'
dy
-=1
dt
'
dz =0
dt
.
(6.2)
This leads to the parametric representation
x=X{s,t)=s+ct,
y= Y{s,t)=t,
z=Z{s,t)=h{s)
(6.3)
for the integral surface. Eliminating s, t we find for the solution of the
initial-value problem (6.1a, b) the representation
z=h{x-ct)
in agreement with (3.3).
14
(6.4)
6 Examples
(2) Euler's P.D.E. for a homogeneous function u(xl, ... ,xn ):
(a = const.=t=O).
(6.5)
Since equation (6.5) is singular at the origin (the J defined by (5.24) cannot
be different from 0) we postulate the initial-value problem
U(XI, ... ,Xn _ l , 1)= h(xl, ... ,xn _ l )
corresponding to a curve
r
(6.6)
given by
fori=I, ... ,n-l
for i= n
(6.7)
z = h(sl' ... ,Sn-I).
Solving the characteristic differential equations
dXi
(j[=Xi
(6.8)
fori=I, ... ,n
dz
dt =az
leads to
for i= 1, ... ,n-l
.for i= n
(6.9)
and thus to
(6.10)
The solution
U
satisfies the functional equation
u(Ax I ,·." Axn) = i\ "u (XI' ... , xn)
(6.11)
for any i\ > 0, and thus is a homogeneous function of degree a.
For a < 0 the solutions of (6.5) generally become singular at the origin.
More precisely the only solution U of (6.5) of class C I in a neighborhood of
the origin is u=O. For along any ray
i= 1, ... ,n
from the origin referred to a parameter t we have by (6.5)
du
-d
t
~
= £.J
k= I
CkUXk ( clt, ... ,cnt)
a
= -u.
t
Hence ut- a is constant along the ray, and thus u tends to 00 for t~O,
unless u vanishes identically along the ray. We have here an example of a
P.D.E. that has only a single solution if we restrict the domain of the
solution to be a set containing the origin.
15
1 The single first-order equation
(3) The solution u = u(x,y) of the quasi-linear equation (see [6])
(6.12)
Uy+uux=O
can be interpreted as a velocity field on the x-axis varying with the time y.
Equation (6.12) then states that every particle has zero acceleration, and
hence constant velocity. Let
(6.13)
u(x,O)= hex)
describe the initial velocity distribution, corresponding to the manifold r
in xyz-space given by
(6.14)
y=O,
z=h(s).
x=s,
The characteristic differential equations
dx
dt =z,
dy
dt =1,
dz =0
dt
(615)
.
combined with the initial condition (6.14) for t=O lead to the parametric
representation
(6.16)
z=h(s)
y=t,
x=s+zt,
for the solution z=u(x,y) of (6.12), (6.13). Eliminating s,t from (6.16)
yields the implicit equation
u=h(x-uy)
(6.17)
for u as a function of x,y. (Notice the analogy to (6.4)!)
The characteristic (projection) Cs in the xy-plane passing through the
point (s,O) is the line
x=s+h(s)y
(6.l8a)
along which u has the constant value
u=h(s).
(6.18b)
Physically (6.18a) for a fixed s represents the path of the particle located at
x = s at the time t = O. Now two characteristics CS1 and CS2 intersect at a
point (x,y) with
(6.19)
If the y in (6.19) is defined, the function u must take the distinct values
h(sl) and h(S2) at (x,y) and hence cannot be univalued. There always exist
positive y of the form (6.19), unless h(s) is a nondecreasing function of s.
For all other h(s) the solution u(x,y) becomes singular for some positivey.
(Physically a particle with a higher velocity will eventually collide with one
ahead of it having a lower velocity). In particular, u is bound to become
singular if the initial velocity distribution h has compact support, except in
the trivial case where h(s)=O. The nature of the singularity becomes
clearer when we follow the values of the derivative uAx,y) along the
16
6 Examples
characteristic (6.18a). We find from (6.17) that
h'(s)
u=
x
Hence for h'(s) <0 we find that
I +h'(s)y
(6.20)
.
becomes infinite at the positive time
-I
(6.21)
y= h'(s)"
Ux
The smallest y for which this happens corresponds to the value s = So at
which h'(s) has a minimum. At the time T= -1/h'(sJ the solution u
experiences a "gradient catastrophe" or "blow-up:' There cannot exist a
strict solution u of class C 1 beyond the time T. This type of behavior is
typical for a nonlinear partial differential equation.
It is possible, however, to define weak solutions of (6.12), (6.13) which
exist beyond the time T. For that purpose (6.12) has to be given a meaning
for a wider class of functions u that do not necessarily lie in C 1 or even are
continuous. We can write (6.12) in divergence form
aR(u) + as(u) =0
ay
ax
'
(6.22)
where R(u), S(u) are any functions for which
S'(u)= uR'(u).
Relation (6.22) implies for any a,b,y the "conservation law"
d
0= dy
fba R (u(x,y»dx+ S(u(b,y»- S(u(a,y».
(6.23)
(6.24)
Conversely (6.22) follows from (6.24) for any u Eel. Now (6.24) makes
sense for more general u and can serve to define "weak" solutions of
(6.22). In particular we consider the case where u is a C I-solution of (6.22)
in each of two regions in the xy-plane separated by a curve x = ~(y), across
which the value of u shall undergo a jump ("shock"). Denoting the limits
of u from the left and right respectively by u- and u+, we find from (6.24)
for a<~(y)< b
O=S(u(b,y»-S(u(a,y»+
~ (iIiR(u)dx+ ~bR(U)dx)
= S(u(b,y»- S(u(a,y»+f R (u-)-f R (u+)
_jliaS(u) dx- [baS(U) dx
ax
ax
Ii
= -(R (u+)- R (u-»f - S(u-)+ S(u+).
a
Hence we find the relation ("shock condition")
d~
dy =
S(u+)- S(u-)
R(u+)-R(u-)
(6.25)
17
I
The single first-order equation
connecting the speed of propagation dV dy of the discontinuity with the
amounts by which Rand S jump. We observe that (6.25) depends not only
on the original partial differential equation (6.12) but also on our choice of
the functions R ( u), S (u) satisfying (6.23).
PROBLEMS
1. Solve the following initial-value problems:
(a) ux+~=U2, u(x,O)=h(x)
(b) ~=xuux' u(x,O)=x
(Answer: x=ue-YU implicitly.)
(c) xux+y~+uz=u, u(x,y,O)=h(x,y)
(d) x~-yux=u, u(x,O)=h(x)
(Answer: u=hCV x 2+y2 )earctan<y/x).)
2. (Picone). Let u be a solution of
a(x,y)ux+b(x,y)Uy= -u
of class C l in the closed unit disk U in the xy-plane. Let a(x,y)x+b(x,y)y >0
on the boundary of U. Prove that u vanishes identically. (Hint:· Show that
maxu,,;O, minu~O.)
!l
!l
3. Let u be a C I-solution of (6.12) in each of two regions separated by a curve
x = €(y). Let u be continuous, but Ux have a jump discontinuity on the curve.
Prove that
d€
dy =u
and hence that the curve is a characteristic. (Hint: By (6.12)
(~+
-:- ~-) + u(ux+ - u;) =0.
Moreover u(€(y ),y) and (d/ dy )u(€(y ),y) are continuous on the curve.)
4. Show that the function u(x,y) defined for y
~
0 by
u= -Hy+ \l3x+y2) for4x+y2>0
u=o for4x+y2<0
is a weak solution of (6.22) for the choice R(u)=u,S(u)=tu2.
S. Define a weak solution u(x,y) of (6.22) as a function for which the relation
(6.26)
holds for any function </>(x,y) of class qr> (Relation (6.26) follows formally from
(6.22) by integration by parts.) Show that this definition of weak solution also
leads to the jump condition (6.25).
6. Show that the solution u of the quasi-linear partial differential equation
~+a(u)ux=O
18
(6.27)
7 The General First-order Equation for a Function of Two Variables
with initial condition u(x,O)=h(x) is given implicitly by
u=h(x-a(u)y)
(6.28)
Show that the solution becomes singular for some positive y, unless a(h(s» is a
nondecreasing function of s.
7. The General First-order Equation for a Function
of Two Variables
The general first-order partial differential equation for a function z =
u(x,y) has the form
F(x,y,z,p,q) =0,
(7.1)
where p=ux,q=Uy. We assume that F where considered has continuous
second derivatives with respect to its arguments x,y,z,p,q. Surprisingly
enough the problem of solving the general equation (7.1) reduces to that of
solving a system of ordinary differential equations. This reduction is
suggested by the geometric interpretation of (7.1) as a condition on the
integral surface z=u(x,y) in xyz-space determined by a solution u(x,y).
The geometry here is more involved than in the quasi-linear case where we
were concerned principally with integral curves. We shall have to deal with
more complicated geometric objects, called "strips."
Equation (7.1) can be viewed as a relation between the coordinates
(x,y,z) of a point on an integral surface and the direction of the normal of
the integral surface at that point, described by the direction numbers
p,q, -1. An integral surface passing through a given point Po=(xo,yo,zo)
must have a tangent plane
z-zo=p(x-xo)+q(y-yo)
(7.2)
for which the direction numbers (p,q, -1) of the normal satisfy
F(xo,yo,zo,p,q) =0.
(7.3)
Thus the differential equation restricts the possible tangent planes of an
integral surface through Po to a one-parameter family. In general* such a
one-parameter family of planes through Po can be expected to envelop a
cone with vertex Po, called the Monge cone at Po (Figure 1.3). Each
possible tangent plane touches the Monge cone along a certain generator.
In this way the partial differential equation (7.1) defines afield of cones. A •
surface z = u (x,y) is an integral surface if at each of its points Po it touches
the cone with vertex Po. (See Figure 1.4.) In that case the generator along
which the tangent plane touches the cone defines a direction on the
surface. These "characteristic" directions are the key to the whole theory
of integration of (7.1). In the special case of a quasi-linear equation (4.2)
the Monge cone at Po degenerates into the line with direction (a,b,c)
through Po.
* For the present heuristic considerations we dispense with rigor.
19
1 The single first-order equation
Figure 1.3
Figure 1.4
The central notion here is that of the envelope of a family of surfaces SA
z= G(X,y,A)
(7.4)
depending on a parameter A. We combine (7.4) with the equation
0= GA (x,y, A).
(7.5)
For fixed A equations (7.4), (7.5) determine a curve 'lA. The envelope is the
union of these curves. Its explicit equation is obtained by solving (7.5) for
A in the form A= g(x,y) and substitution into (7.4):
z=G(x,y,g(x,y»).
(7.6)
The envelope E touches the surface SA along the curve YA• For in a point
(x,y,z) of YA we have g(X,y)=A, so that (7.6) holds; moreover by (7.5) the
direction numbers of the normal of E are (Gx + GAgx , Gy + GAgy ' -1)=
(Gx,Gy ' -1), the same as for SA. It is often advantageous to write equations
(7.4), (7.5) for a fixed A in the form of differential equations
0= G7I.x dx + GAY dy
(7.7)
satisfied along the curve YA•
Now for fixed xo,yo,zo, equation (7.2) defines a one-parameter family of
planes, where we can choose p as the parameter and think of q as
expressed in terms of p from (7.3). By (7.7) the generator along which the
plane touches the Monge cone satisfies the equations
dq
(7.8)
dz=pdx+qdy,
O=dx+ dp dy.
Since by (7.3)
(7.9)
20
7 The General First-order Equation for a Function of Two Variables
the direction of the generator is given by
dz=pdx+qdy,
dx=dy
F;, Fq
•
(7.10)
On a known integral surface S: z=u(x,y) equations (7.10) define a
direction field, since F;,(x,y,u,ux,zs,), Fq(x,y,u,ux'zs,) are then also known
functions of x,y. We define the characteristic curves belonging to the
integral surface S as those fitting the direction field. Using a suitable curve
parameter t, the characteristic curves on S are given by the system of
ordinary differential equations
dy
dx
(7.10a)
dt = Fq(x,y,z,p,q)
dt = F;, (x,y,z,p,q),
dz
dt
= pF;, (x,y,z,p,q) + qFq (x,y,z,p,q),
(7.10b)
where
z=u(x,y),
p=uxCx,y),
q= zs,(x,y).
(7.11)
It is clear that for a quasi-linear equation (4.2) relations (7.10a,b) reduce
to the characteristic differential equations (4.4). The main difference in the
present more general case is that without the use of (7.11), that is, without
the knowledge of the integral surface S, equations (7.lOa,b) form an
underdetermined system for the five functions x,y,z,p,q of t. However, the
system is easily completed by two further equations. Partial differentiations
with respect to x,y of the P.D.E.
F(x,y, u(x,y), uxCx,y), zs, (x,y» =0
(7.12)
furnish the relations
Fx + ux~ + uxxF;, + uxyFq=0
(7.13a)
F;, + zs,Fz + uxyFp + ~Fq=O.
(7.13b)
Then along a characteristic curve on S we have by (7.11), (7.lOa)
dx
dy
dt = uxx dt + Uxy dt = uxxF;, + uxyFq = - Fx - uxFz
dp
with a similar relation for dqj dt. Writing these relations in the form
dp
= - FxCx,y,z,p,q) - pFzCx,y,z,p,q)
(7.14a)
dq
dt = -F;,(x,y,z,p,q)-qFzCx,y,z,p,q)
(7.14b)
dt
relations (7.lOa,b), (7.14a,b) constitute an autonomous system of five
ordinary differential equations for the five functions x,y,z,p,q of t, which
does not require knowledge of the integral surface z = u(x,y) for its
formulation.
21
1 The single first-order equation
The expression F(x,y,z,p,q) is an "integral" of the system, that is, F is
constant along any solution, since by (7.lOa,b), (7.14a,b)
~
~
~
~
~
~
dt=~~+Ey~+~~+~~+~~
=Fx~+EyFq+Fz(p~+qFq)+~( -Fx-p~)+Fq( -Ey-qFz)=O.
Hence along any trajectory of the system we have F=O for all t, if F=O
for some particular t. We refer to the system of five ordinary differential
equations (7.lOa,b), (7.14a,b), together with the dependent relation
F(x,y,z,p,q)=O
(7.15)
as the characteristic equations.
A solution of the characteristic equations is a set of five functions
x(t),y(t),z(t),p(t),q(t). Generally we call a quintuple (x,y,z,p,q) a plane
element and interpret it geometrically as consisting of a point (x,y,z) and
of a plane through the point with equation
r - z=p(~- x)+ q('Ij - y)
(7.16)
in running coordinates ~,'Ij,r. Then p,q, -1 are direction numbers of the
normal of the plane. An element is called characteristic if it satisfies (7.15).
A one-parameter family of elements (x(t),y(t),z(t),p(t),q(t» is called a
strip if the elements are tangent to the curve formed by the points
(x(t),y(t),z(t», the support of the strip. For that to be the case, the strip
condition
(7.17)
has to be satisfied. A solution of the characteristic equations will be called
a characteristic strip. (Here the strip condition holds in consequence of
relations (7.lOa,b).) (See Figure 1.5.)
A surface z=u(x,y) referred to parameters s,t can be thought of as
consisting of a two-parameter family of elements (x(s, t),
z(s,t),p(s,t), q(s,t» formed by the points of the surface and the corresponding tangent planes. Not every two-parameter family of elements.
forms a surface. It is necessary again that
dz=pdx+q~
Figure 1.5
22
(7.18)
8 The Cauchy Problem
holds along the family, i.e., that the strip conditions
az _ ax + ay
as -Pas q as'
az _
ax
at -Pat
+
ay
qat
()
7.19
are satisfied. Visualizing an element as a small piece of plane attached to a
point, the elements of a two-parameter family belonging to a surface have
to fit together smoothly somewhat like scales on a fish.
As a solution of a system of ordinary differential equations a characteristic strip is determined uniquely by anyone of its elements. If that
element of the strip consists of a point P of an integral surface S and of the
tangent plane to S at P, then the strip is made up of the characteristic
curve of S through the point P and of the tangent planes of S along that
curve. If another integral surface touches S in the point P then that surface
will also touch S all along the characteristic curve.
Equations (7.lOa,b), (7.14a,b) describe a "law of propagation" of characteristic elements or of tangent planes on an integral surface along
characteristic curves. A lower-dimensional geometric interpretation often is
useful. We associate with (x,y,z,p,q) a "line element" in the xy-plane
consisting of the point (x,y) and the line
O=p(g-X)+q(l1-Y)
(7.20)
in running coordinates g,l1. The line (7.20) is the level line r = z of the
plane (7.16). The geometric line element does not determine x,y,p,q
uniquely, since p and q can be replaced by any proportional numbers. A
characteristic strip gives rise to a family of line elements. Using z instead
of t as a parameter, the line elements "propagate" for varying z in a
definite way described by the equations
F;,
dx
dz
= pF;, + qFq ,
dp
dt
Fx+pFz
= - pF;,+qFq '
dy
dt
Fq
= pF;, + qFq
dq
~+qFz
dt = - pF;, + qFq .
(7.2Ia)
(7.2Ib)
We can start out with an initial line element (xo,yo,Po,qo) for z = Zo (in
which Po, % are to be replaced by proportional quantities so that
F(xo,yo,zo,Po,%) =0
(7.22)
holds). With changing z the line element moves along the curve x = x(t),
y = yet) (called a "ray"). The line elements are not tangent to the ray (no
"strip condition"), since generally
dx
dy
p dt +q dt =pF;,+qFq=l=O.
8. The Cauchy Problem
The Cauchy problem for (7.1) consists in passing an integral surface
through an "arbitrary" initial curve r given parametrically in the form
x=j(s),
y=g(s),
z=h(s).
(8.1)
23
1 The single first-order equation
This will be achieved by passing suitable characteristic strips through r.
We assume thatj,g,h are of class C 1 for s near a value So corresponding to
a point
(S.la)
We first have to complete r into a strip consisting of characteristic
elements. That is, we have to find functions
p=</>(s),
(S.2)
q=t/l{s)
such that
h'(s) = </>(s)f'(s) + I/I(s) g'(s)
(S.3a)
F(j(s),g(s),h(s),</>(s), 1/1 (s)) =0.
(S.3b)
Since equation (S.3b) is nonlinear there may be one, or several, or no
solution (</>,1/1) of (S.3a,b). We assume that we are given a special solution
Po>qo of
(S.4)
such that
d= j'(so)Fq (xo,yo,zo,Po, qo) - g'(so).F;. (xo>yo,zo,Po,qoh=O.
(S.5)
That is, we have found a characteristic plane tangent to r at Po such that
the generator of contact between the plane and the Monge cone at Po has
a different projection onto the xy-plane than the tangent to r at Po. By the
implicit function theorem there exist then unique functions <P(s), 1/1 (s) of
class C 1 near so' satisfying (S.3a,b) and reducing to Po>qo for s=so.
Through each element (f(s),g(s),h(s),<p(s),I/I(s» we now pass the characteristic strip reducing to that element for 1=0. In this way we find five
functions
x=X(s,/),
y= Y(s,/),
z=Z(s,t),
p=P(s,t),
q=Q(s,t)
(S.6)
defined for Is-sol and It I sufficiently small, which satisfy for fixed s the
differential equations (7.1Oa,b), (7.14a,b) as functions of t, and for s=o
reduce respectively to j(s),g(s), h(s), <p(s), 1/1 (s). The relation
F(X, Y,Z,P,Q)=O
(S.7)
holds identically in s,t, since it holds for t=O by (S.3b).
From what preceded it is clear that if there exists an integral surface S
through r containing the element (xo,yo,zo,Po>qo), then that surface must
be the union of the supports of the characteristic strips we constructed. In
particular, the first three equations in (S.6) must constitute a parametric
representation for S.
Conversely we shall prove that (S.6) represents a solution of our Cauchy
problem in parametric form in a neighborhood of the point Po. First of all
we can solve the first two equations (S.6) for s, t in terms of x,y for (x,y)
24
8 The Cauchy Problem
near (xo,Yo). For by (8.la)
Xo= f(so)=X(so,O),
yo=g(so) = Y(so,O)
and we have a nonvanishing Jacobian at s=so, 1=0:
a (x,y) /Xs
a (s,1 ) = xt
(8.7a)
Substituting into the third equation (8.6) yields an explicit equation z =
u(x,y) for a surface passing through r. By (8.7) this will be an integral
surface, if we can show that the p,q defined by the last two equations (8.6)
are identical with Ux and ~. Now from the first three equations in (8.6) we
can determine Ux ' zs, uniquely in terms of s,1 by the chain rule which yields
the relations
Zs = uxXs + zs, Ys'
Zt = uxXt + uy
y,
(8.8)
We have proved that
ux=P(s,/),
zs,=Q(s,/),
(8.9)
if we can verify the identities
Zs = PXs + QYs,
Zt = PXt + QY,.
(8.10)
These equations just express that the two-parameter family of elements
(8.6) belongs to a surface. The second equation in (8.10) is a consequence
of the characteristic differential equations (7.lOa,b) satisfied by X, Y,Z,
P, Q, as functions of I. To verify the first equation we introduce the
expression
(8.11)
Here
A (s,O) = h' -cf>f' - t[;g' =0
(8.12)
by (8.3a). Moreover, making use of the characteristic equations for
X, Y,Z,P,Q, we have
At = Zst - PtXs - Qt Ys - PXst - QYst
a
= as (Zt - PXt - QYt)+PsXt + QsYt - Qt Ys - PtXs
=PsF;,+ QsFq+Xs(Fx+ FzP)+ Ys(r;. + FzQ)
=
aF
as
-~(Zs-PXs- QYs )= -~A.
(8.13)
But then, by integration,
A (S,/)=A(s,O)exp( - fotFzdl) =0
because of (8.12). This completes the local existence proof for a solution of
the Cauchy problem under the assumption that we have a solution Po, qo of
(8.4), (8.5).
25
1 The single first-order equation
The same methods apply to first-order equations in more independent
variables. The general equation for a function u(x" ... ,xn ) has the form
F(x" ... ,xn,z,p" ... ,Pn)=O,
(S.14)
where z = u, Pj = Ux .' The Cauchy problem here consists in finding an
integral surface iIi. x, ... xnz-space that passes through an (n -1)dimensional manifold r given parametrically by
z=h(s",,,,sn_')'
X j = /;(s",,,,sn_') for i= l, ... ,n.
This is achieved as before by passing through each point P of r a
characteristic strip tangent to r at P. We first complete r into a strip by
finding functions pj = q,j(Sj, ... ,sn-') for which
ah ~ afk
a-= ~q,kaSj k='
Si
.
forl=l, ... ,n-l
(S.14a)
(S.14b)
in analogy to (S.3a,b). A characteristic strip here is a set of "elements"
(x" ... ,xn'z,p" ... ,Pn) depending on a parameter t that satisfies in addition
to (S.14) the system of ordinary differential equations
dXj
di = F;",
dz
dt =
tipj
dt = - Fx , - FzPj
for i= 1, . .. ,n
n
.~ PiF;,,'
(S.15a)
(8.15b)
1= ,
For existence assumption (8.7a) here has to be replaced by
a=
aft
as,
afn
as,
aft
asn_,
afn
asn_,
F;"
F;,n
~O
at one point of r.
An instructive example is given by the equation
C2(p2 + q2) = 1
(8.l5c)
(8.16)
(c = const. > 0), which arises in geometric optics. There the level lines of a
solution u are interpreted as "wave fronts," marking the location to which
light has spread. (We shall obtain the same lines as lines of discontinuity
for solutions of the wave equation.) Geometrically (8.16) states that tangent planes of an integral surface make a fixed angle O=arctanc with the
z-axis. The Monge cone enveloped by possible tangent planes through a
point is then a circular cone with opening angle 20.
26
8 The Cauchy Problem
For convenience we write (S.16) as
F=Hc7;2+ C2q2-1)=0
(S.17)
leading to the characteristic equations
dx =c2..
P,
dt
~~
dy =c2q
dt
'
= C2(p2+ q2)= I,
dp
dt
dq
dt
-=-=0
.
(S.IS)
A given initial curve
r:
z=h(s)
(S.19)
is completed into a strip by choosing p=cp(s),q=l/I(s) according to the
equations
x=f(s),
y=g(s),
h'(s) = cp(s )f'(s) + l/I(s) g'(s),
(S.20)
2
These equations have no real solution when j'2 + g,2 < c h'2, that is, when r
forms an angle less than 0 with the z-axis (f is "timelike"). For a
"spacelike" r with j'2 + g,2 > c2h,2 we can solve (S.20) in two different
ways, giving rise to two different solutions of the Cauchy problem.
Of special interest is the case where r is the intersection
x=f(s),
y=g(s),
Z=O
(S.21)
of the integral surface z = u(x,y) with the xy-plane. For this space-like r
there are two solutions (cp(s),l/I(s» of
O=cpf' + l/Ig',
cp2+l/I 2=C- 2,
(S.22)
differing only in sign. The characteristic strips with initial elements on r
are given by
(S.23a)
y = g(s) + c2tl/1(s),
x = f(s) + c2tcp(s),
(S.23b)
Z = t,
P =cp(s),
q=l/I(s).
We represent the integral surface by its level lines in the xy-plane
y/:
u(x,y)=z=const.=t
(S.24)
interpreted as a "wave front" moving with the "time" t. Equations (S.23a)
for fixed t give the curve y/ referred to the parameter s. We think of y/ as
made up of line elements (x,y,p,q), i.e., of points (x,y) and corresponding
tangents
(S.25)
p(~- x) + q(l1- y) =0.
The elements propagate individually according to (S.23a,b) for fixed s. The
point (x,y) moves along the ray (S.23a) with constant speed
(: r+(: r
=C~cp2+l/I2
=c.
(S.26)
The ray is the straight line of direction (cp,l/I)=(p,q), hence coincides at
27
1 The single first-order equation
each point with the normal to the level line 'Yt. We see that the wave fronts
'Yt form a family of curves with common normals ("parallel" curves). Here
'Yt can also be obtained from 'Yo by laying off a fixed distance ct along each
normal of 'Yo. Thus 'Yt is also the curve of constant "normal distance" ct
from 'Yo' (The choice between "interior" or "exterior" normal directions
corresponds to the choice between the two solutions of the Cauchy
problem.)
9. Solutions Generated as Envelopes
The envelope S of a family of integral surfaces SA of (7.1) with equation
z= G{X,y,A)
(9.1)
is again an integral surface. This is obvious from the geometric interpretation of (7.1) and the fact that every tangent plane of the envelope is a
tangent plane of one of the surfaces of the family. Here S touches SA along
the curve given by the two equations
z= G{X,y,A),
0= GA {x,y, A).
(9.2)
Since integral surfaces can touch only along characteristic strips, we see
that completing equations (9.2) for fixed A by
p=GAX,y,A),
q=Gy{X,y,A)
(9.3)
describes a characteristic strip. We obtain its parametric representation
with (x,y,z,p,q) as functions of some parameter t by solving the equation
0= GA (x,y, A) in some parametric form x = x (t),y = y(t) and substituting
into the other equations. In this way a one-parameter family of solutions of
the partial differential equation (7.1) yields a further solution and a
one-parameter family of characteristic strips, purely by elimination.
Using this observation we can obtain the "general" solution of (7.1) and
the "general" characteristic strip from any special two-parameter family
u= G{X,y,A,P.)
(9.4)
of solutions of (7.1). Many different one-parameter families are obtained
from (9.4) by substituting for p. any function p. = w(A). Here for a particular
A and a suitable function w the values of p. = w(A) and v = W'(A) are
arbitrary. It follows that for general A, p., v the equations
(9.4a)
0= GA (X,y,A,P.):t- vG,. (X,y,A,P.)
z = G (x,y, A, p.),
(9.4b)
q= Gy (X,y,A,P.)
p = GAX,y,A,P.),
describe a characteristic strip. We obtain in this way the "general"
characteristic strip containing an arbitrary characteristic element
(xo,yepzepPo,qo)' We only have to determine A,P.,V in such a way that
equations (9.4a, b) hold for x,y,z,p,q replaced by xo,Yo, Zo,Pep qo· These are
essentially only 3 conditions for A,P., v since the equation F(x,y,z,p,q) =0
holds for all elements determined from (9.4a, b) as well as for the
prescribed characteristic element.
28
9 Solutions Generated as Envelopes
The Cauchy problem of finding an integral surface of (7.1) passing
through a curve r can be solved directly with a knowledge of the
two-parameter family (9.4). We only have to find a one-parameter subfamily of integral surfaces that touch r and then form their envelope. For
r given by (8.1) this amounts to finding functions A=A(S), p.=p.(s) such
that the relations
h = F(j,g,A,P.),
(9.5)
h'= GAf,g,A,p.)f' + Gy (j,g,A,P.)g'
hold identically in s. The solution is then determined by eliminating S from
the equations
u= G (X,y,A,P.),
0= GA (x,y,A,p.)N + G,. (X,y,A,P.) p.'.
(9.6)
Of special interest is the limiting case, when r degenerates into a point
(xo,Yo,zO>. Taking the solutions (9.4) for which A,P. are such that
Zo= G(Xo,yo,A,P.)
and forming their envelope we obtain an integral surface with a conical
singularity at (xo,yo,zo), called the "conoid" solution. The solution of the
general Cauchy problem can be obtained by taking the envelope of the
conoids that have their singularities on the prescribed curve r.
Any two-parameter family of functions (9.4) determines a first-order
partial differential equation (7.1) of which these functions are solutions.
We only have to eliminate the parameters A, p. between the equations
u=G(X,yA,P.),
ux=GAx,y,A,P.),
~=Gy(x,y,A,p.).
(9.7)
For the resulting partial differential equation we can immediately find all
characteristic strips and solve the Cauchy problem.
An example is furnished by the equation
c2(u;+u;)=1
(9.8)
consider earlier. The equation has the special two-parameter family of
linear solutions
u= G (X,y,A,P.) = c-1(XCOSA+ ysinp.)+ p.
(9.9)
representing planes in xyz-space forming an angle O=arctanc with the
z-axis. The general integral surface z = u(x,y) is the envelope of a oneparameter subfamily of the planes (9.9). The conoid with "vertex" or
singular point (xo,yo,zo), here identical with the Monge cone, is the
envelope of the planes (9.9) passing through (xo,yo,zo), i.e., of the planes
z - Zo= c-1(x- Xo)COSA+(y - Yo)SinA).
(9.10)
The conoid thus is the circular cone
(9.11)
c2(z - zo)2 = (x - xoi + (y - YO)2.
We can find the solution (9.8) passing through an initial curve "Yo in the
xy-plane by forming the envelope of the conoids with vertex on "Yo. Since
the level line z=const. of the conoid with vertex (xo,yo,O) is a circle of
radius ct and center (xo,yo), we obtain the level line "YI of the envelope by
29
I The single first-order equation
forming the envelopes of the circles of radius ct with centers on 'Yo' This
agrees with the earlier representation of 'Yt as curve of normal distance ct
from 'Yo. It corresponds to Huygens's generation of the wave front 'Yt as the
envelope of "circular" waves issuing from the original wave front 'Yo.
PROBLEMS
1. For the equation
u;+u;=u 2
find (a) the characteristic strips; (b) the integral surfaces passing through the
circle
x=coss,
y=sins,
z= I
[Answer: z=exp[±(I-Vx2+y2)]]; (c) the integral surfaces through the line
x=s,y=O, z= 1 [Answer: u=exp(±y)].
2. For the equation
Uy =
u;
(a) find the solution with u(x,0)=2X 3/ 2 [Answer: u=2x 3/2(1-27y)-1/2]; (b)
prove that every solution regular for all x,y is linear.
3. For the equation
u = xUx + yUy +
find a solution with u(x, 0) =
HI - X2).
Hu; + U;)
4. Given the family of spheres of radius I with centers in the xy-plane
u= G(x,y,lI.,p.)=YI-(x-lI.i-(y- pi
,
find the first-order partial differential equation they satisfy. Find all characteristic strips and give a geometric description. Find the conoid solution with vertex
(O,O,D. Find the integral surfaces through the line x=s,y=O, z=t.
5. (Characteristics as extremals of a variational problem). Consider for a given
function H (Xl'''' ,Xm t'PI>'" ,Pn) the partial differential equation
F=
au +H (Xl,· .. ,xn,t, ax!
au , ... , aXn
au ) =0
at
(Hamilton-Jacobi equation) for U=U(XI, ... ,Xmt). Obtain the characteristic
equations in the form
dxi
dpi
-=-H
(ji=Hp"
dt
x,'
Setting dXi / dt = Vi' du / dt = L we use the first n + I equations to express L as a
function of XI,. .. ,xmt,vJ, ... ,vn. Show then that
4,=Pi'
which implies that
d
dt 4 , -Lx,=O.
These are the Euler-Lagrange equations for an extremal of the functional
f L(xl> ... ,xmt,dxt/dt, ... ,dxn/dt)dt.
30
Second-order equations:
hyperbolic equations for
functions of two
independent variables *
2
1. Characteristics for Linear and Quasi-linear
Second-order Equationst
We start with the general quasi-linear second-order equation for a function
u(x,y):
(1.1)
auxx + 2buxy -+ CUyy = d,
where a,b,c,d depend on x,y,u,ux,lly. Here the Cauchy problem consists of
finding a solution u of (1.1) with given (compatible) values of u, ux' uy on a
curve y in the xy-p1ane. Thus, for y given parametrically by
x= f(s),
y =g(s),
(1.2)
we prescribe on y
u=h(s),.
ux=cp(s),
(1.3)
The values of any function v(x,y) and of its first derivatives vAv,y, v/x,y)
along the curve yare connected by the compatibility condition ("strip
condition")
dv
(1.4)
ds =vJ'(s)+Vyg'(s)
which follows by differentiating v(f(s),g(s» with respect to s. Applied to
the solution u of the Cauchy problem this implies the identity
h'(s) =cp(s )f'(s) + 1{;(s )g'(s)
(1.5)
between the Cauchy data. Thus no more than two of the functions h, cp, 1{;
* ([7], [13], [6])
t([22D
31
2 Second-order equations
can be prescribed arbitrarily. Instead we might give on y the values of u
and of its normal derivative:
- u g' +Uy./'
x
X(s).
(1.6)
u=h(s),
YJ'2+ g ,2
Compatibility conditions also hold for the higher partial derivatives of
any function on y. Thus taking v = Ux or v = Uy we find that
du
;
= uxxf'(s) + uxyg'(s),
dUy
di= ux;1'(s) + lIyyg'(s).
(1.7)
Similar relations are valid for the s-derivatives of Uxx' uxy' Uyy' etc.
If now u(x,y) is a solution of (Ll), (1.3) we have the three linear
equations
auxx + 2buxy
+ cUyy = d
(1.8a)
cp'
(1.8b)
f' Uxy + g' Uyy = \[;'
(1.8c)
f' Uxx + g' Uxy
=
for the values of uxx,uxy,Uyy along y, with coefficients that are known
functions of s. These determine uxx,uxy,Uyy uniquely unless
J'
g'
A= 0
a
J'
2b
0
g'
c
= ag,2 - 2bJ' g' + CJ'2 =
o.
(1.9)
We call the "initial" curve y characteristic (with respect to the differential
equation and data), if A=O along y, noncharacteristic if A*O along y.
Along a noncharacteristic curve the Cauchy data uniquely determine the
second derivatives of u on y. As a matter of fact, we can then also find
successively the values of all higher derivatives of u on y, as far as they
~xist. We obtain, e.g., three linear equations with determinant A for
"xxx' uxxy , uxyy by differentiating (Ll) partially with respect to x, and using
the two equations obtained from (1.4) for v = Uxx and v = uxy; the
c.oefficients in the three equtions only involve the values of u and its first
and second derivatives, known already. Obtaining in this way the values of
all derivatives of u in some particular point (xo,yo) on y, we could write
down a formal power series for the solution of the Cauchy problem in
terms of powers of x - xo,y - Yo. It would be an actual representation of
the solution u in a neighborhood of (xo,Yo), if u were known to be analytic.
This procedure will be legitimized in the case of analytic Cauchy data,
when we discuss the Cauchy-Kowalewski theorem below.
In the case of a characteristic initial curve y, equations (l.8a, b, c) are
inconsistent, unless additional identities are satisfied by the data. Hence
the Cauchy problem with Cauchy data prescribed on a characteristic curve
32
2 Propagation of Singularities
generally has no solution. Write condition (1.9) for a characteristic curve
as
a dy2 - 2b dx dy + Cdx 2= O.
We can solve (LlO) for dy / dx in the form
(LlO)
dy = b± Yb 2 -ac
(1.11)
dx
a
When the characteristic curve y is given implicitly by an equation q,(x,y) =
const., we have CPx dx + </y dy = 0 along y so that (Ll 0) reduces to the
equation
acp; + 2bcpx</y + ccp; = o.
(Ll2)
Relation (LlI) is an ordinary differential equation for y provided a,b,c
are known functions of x,y. This is the case when either a fixed solution
u= u(x,y) of (Ll) is considered, or when the equation (Ll) is linear, that
is,
a=a(x,y),
b=b(x,y),
c=c(x,y),
d= - 2D(x,y )ux - 2E(x,y)~ - F(x,y)u - G(x,y).
(1.13)
The equation (1.1) is called elliptic if ac - b 2 > 0, hyperbolic if ac - b2 < 0,
and parabolic if ac - b 2 = O. Restricting ourselves to the case of real
variables, we observe that corresponding to the choice of ± in (LlI) there
are two families of characteristic curves in the hyperbolic case, one in the
parabolic case, and none in the elliptic case. We note, though, that in the
nonlinear case, ,"type" (elliptic, parabolic, hyperbolic) is not determined by
the differential equation, but can depend on the individual solution, and
even for a linear equation might be different in different regions of the
plane.
2. Propagation of Singularities
The characteristic curves are closely associated with the propagation of
certain types of singularities. Along a noncharacteristic curve the Cauchy
data uniquely determine the second derivatives of a solution. One approach to defining "generalized" solutions of (1.1), not necessarily of class
C 2, consists of considering solutions of class C l with second derivatives
that have jump discontinuities along a curve y. More precisely. we assume
that we have a certain region in the xy-plane divided by a curve y into two
portions I and II. There shall be two solutions uI and uII of (1.1)
respectively defined and of class C 2 in the closed regions I and II. uI and
uII together define a function u in the union of I and II with discontinuities
along y. The resulting u, pieced together ordinarily cannot be considered a
"generalized" solution of (1.1), unless (1.1) in some generalized sense still
holds on y. This requires certain tramition conditiom along y. The simplest
to consider here is the case where the resulting u is required to be of class
33
2 Second-order equations
C I, and hence the functions uI and u ll as well as their first derivatives
coincide along y. If also the second derivatives coincide, u actually is a
"strict" solution of class C 2 • Of interest, therefore, is the case where the
second derivatives of uI and ull are not the same on y. Since however, by
assumption u I and u ll have the same Cauchy data along y, a discontinuity
in the second derivatives of u can only occur if y is a characteristic curve.
We analyze the situation in more detail for a linear second-order
equation
0= Lu= a(x,y )uxx +2b(x,y )uxy + c(x,y)Uyy
+2d(x,y)ux +2e(x,y)~ + j(x,y)u
(2.1)
with regular coefficients a,b,c,d,e,j. Let y be a curve in the xy-plane given,
say, by an equation x = <f>(y). (Precise regularity assumptions justifying
what follows can easily be supplied by the reader. These are not relevant
for the present discussion.)
Let Sl be an open set in the xy-plane and y be an arc in. Sl such that
Sl- y consists of two open disjoint sets I and II. We say that a function u
defined in Sl has a jump discontinuity along y, if u = u I in I, u = UIl in II,
where u I is continuous in 1+ y and u ll is continuous in 11+ y. We denote
by
(2.2)
the jump of u at a point (<f>(y),y) of y. Then along y
d
dy [ u] = ( U;I(<f>(y ),y ) - u; (<f>(y ),y ) )<f>'(Y )+ ~I(<f>(y ),y ) - ~ (<f>(y ),y )
= [ux]<f>' + [uy J.
In particular [u] = 0 when u is continuous along y, and thus
(2.3)
(2.4)
We consider now the case where uI(x,y), uII(x,y) are of class C 3 and
satisfy (2.1) respectively in 1+ y and II + y, defining a function u of class
C l in Sl. Then
[ u] = [ ux ] = [ ~ ] = o.
(2.5)
Moreover, subtracting (2.1) formed for u I and u ll along y yields
a[ uxx ] + 2b[ uxy] + c[ Uyy] =0,
(2.6)
since a,b,c,d,e,j are continuous. From (2.5) we find that
[ uxx ]<f>' + [ uxy] = 0,
[ Uxy ]<f>' + [ Uyy] =0.
(2.7)
Hence for a C I-function u the jumps in the second derivatives are not
independent. Knowing, for example, the jump of Uxx' those of uxy, Uyy are
uniquely determined. Setting [Uxx]=A, we have
[ uxx ] = A,
34
[ Uxy ] = - <f>'A,
[ ~y ]
= <f>,2A•
(2.8)
3 The Linear Second-order Equation
For a solution of the differential equation it follows now from (2.6) that
a - 2bcp' + Ccp,2 = 0
(2.9)
unless A=O, that is, unless uE C 2• Recalling that Cp'=dx/ dy, equation (2.9)
asserts that 'Y is a characteristic curve. According to (2.8) the quantity A
measures the "intensity" of the jumps in the second derivatives.
We interpret y as the time, and cp as the point x = cp(y) moving along the
x-axis. For every y we have in u(x,y) a function of x whose second
derivative is discontinuous at the moving point cp(y). The speed dx / dt of
"propagation of the discontinuity" is determined by (2.9).
It is remarkable that the jumps in different points of 'Y are related to
each other. There is a definite law according to which the intensity A of the
jump propagates along 'Y. To find it we derive from (2.8) the relations
A' =
':t =
[uxxx ]cp' + [ uxxy ],
- (CP'A)' = [uxxy ]cp' + [uxyy J.
(2.10)
In addition, differentiating (2.1) formed for uI and uII with respect to x
and forming the jumps on 'Y, we find that
0= a [ uxxx ] + 2b [ uxxy ] + c [ uxyy] + ax [ uxx ]
+ 2bx [ uxy] + cx[ Uyy ] + 2d[ uxx ] +2e[ uxy]
Eliminating the third derivatives, using (2.8) and (2.9), one arrives at the
relation
0= 2( b - ccp')A' + (ax - 2bA' + cA,2 + 2d - 2ecp' - Ccp")A.
(2.11)
This is an ordinary differential equation for the jump intensity A, which
regulates its growth during propagation. If, for example, A=O in one point
of 'Y, it follows that A=O all along 'Y, so that no jump at all occurs.
3. The Linear Second-order Equation
We analyze in more detail the linear second-order equation
auxx+2buxy+c~ +2dux+2~+ ju=O
(3.1)
with coefficients a, b, c, d, e,j depending on x,y.
Introducing new independent variables ~,1/ by the substitution,
~=CP(x,y),
1/=t/I(x,y),
(3.2)
we transform this linear equation into one of the same type,
Lu=A(~,1/)u~E+2B(~,1/)ufrl+'" =0,
(3.3)
35
2 Second-order equations
where,
A = act>; + 2bct>Ay + cct>;
B = act>xo/x + b(ct>xo/y +ct>yl[;x) + c<lyl[;y
(3.4)
C = 01[;; + 2bl[;xo/y + cl[;;
etc.
This suggests, in the hyperbolic case, that one can simplify the differential
equation by introducing the characteristics as new coordinate lines. Let
~=ct>(x,y) = const.,
11 = I[;(x,y) = const.
be the two families of characteristics in the xy-plane, so that both ct> and I[;
satisfy (1.12). By (3.4) this implies that A = 0 and C = 0, and the hyperbolic
equation reduces, after division by B, to the normal form,
ue.., + 2Du~ + 2Eu." + Fu = O.
(3.5)
The new equation has the lines ~ = const. and 11 = const. as characteristic
curves. By a further linear transformation,
X'=~+l1,
(3.6)
Y'=~-l1,
one can also transform (3.5) into the alternate form
u.,,'y'- ux'x' +2D'ux' + 2E'u.", + F'u=O.
(3.7)
In the elliptic case, where ac - b2 > 0, there exist no real characteristics.
We can attempt to find a real transformation (3.2) taking (3.1) into an
equation of the form,
u~~ + u."." + 2Du~ + 2Eu." + Fu =
o.
(3.8)
This means we want to choose ct>(x,y),I[;(x,y), so that A = C, B=O. This
can be achieved by taking for ct>, I[; solutions of the system of equations,
ct>x =
bl[;x + cl[;y
W
'
ct>y = -
01[;x + bl[;y
W
'
(3.9)
where W = Vac - b2 • Eliminating ct> from (3.9) we see that I[; has to be a
solution of the Beltrami equation,
(3.10)
EXAMPLE
(The Tricomi Equation).
Uyy - YUxx=O.
(3.11)
For this equation, ac - b2= - y. Hence for y < 0, ac - b2> 0, and the
equation is elliptic. For y > 0, ac - b 2 is negative and the equation is
hyperbolic. On the x-axis, it is parabolic. (See Figure 2.1.) Here, the
characteristic equation (1.10) reduces to - yt(y2+ dx 2=0 or
dx±Vy t(y=0 fory>O.
36
(3.12)
3 The Linear Second-order Equation
y
o
x
elliptic
Figure 2.1
The characteristic curves in the half plane y
> 0 are therefore
3x ± 2(y )3/2 = const.
(3.13)
The transformation,
(3.14)
~=3x-2y3/2,
reduces the equation to the normal form:
I uE- u"
(3.15)
uE'I-'6 ~-1J =0.
The curves
3x - 2y 3/2 = canst.
are the branches of cubic cUrves having positive slope and -the--curves
3x + 2y 3/2 = canst.
are the symmetric curves with negative slopes. On y = 0, the curves have
cusps with a vertical tangent.
PROBLEMS
1. For the equation of minimal surfaces (2.12), p. 3 of Chapter 1, find
(a) all minimal surfaces of revolution about the z-axis (i.e., u =
x 2 +y2
(b) the differential equations for the (imaginary) characteristic curves.
In/
)
2. Find by power series expansion with respect to y the solution of the initial-value
problem
~=u",,+u
u(x,O)=e",
Uy(x,O) =0.
3. (Legendre transformation, Hodograph method). Let u(x,y) be a solution of a
37
2 Second-order equations
quasi-linear equation of the form
a(ux,uy)uxx +2b(ux ,uy )uxy + c(ux'~)uy'y =0.
Introduce new independent variables
~=uAx,y),
~,1/
and a new unknown function If> by
1/=uy (x,y),
If>=xux+Yuy-u.
(3.16)
Prove that If> as a function of ~,1/ satisfies x = If>~, Y = <Pr, and the linear differential
equation
(3.17)
4. The One-Dimensional Wave Equation
The simplest of all hyperbolic differential equations is the one-dimensional
wave equation
(4.1)
where u is a function of two independent variables x and t and c denotes a
positive constant. The variable x is commonly identified with "position"
and t with "time"; c is a given positive constant. Physically u can represent
the normal displacement of the particles of a vibrating string. Here the
characteristics are the two families of lines x ± ct = constant in the xtplane. Introducing them as coordinates by putting
(4.2)
x+ ct=~,
x-ct=1J,
(4.1) becomes
(4.3)
u~=O.
Assume that the domain of u, as a function of x, t or, equivalently, as a
function of ~,1J is convex. Since (u~)." = 0 it follows that u~ is independent of
1J, say, ul;.= J'm, and then u= Jf'm~+ G(1J). That is,
u=F(~)+ G{1J).
(4.4)
In the original variables we find that u is of the form
(4.5)
Here u E C 2 if and only if F, G, E C 2• Thus the general solution of (4.1) is
obtained by superposition of a solution F(x+ct)=v of vt-cvx=O and of
a solution G(x-ct)=w of wt+cwx=O. This corresponds to the fact that
the differential operator
u=F(x+ct)+ G(x-ct).
(4.6)
can be decomposed into
L= (i._
at
c~)(i.+
c~).
ax
at
ax
(4.7)
Thus the graph of u(x, t) in the xu-plane consists of two waves propagating
without change of shape with velocity c in opposite directions along the
x-axis. (See p. 5.)
38
4 The One-Dimensional Wave Equation
We impose the initial conditions
ut(x,O)=g(x).
(4.8)
For u of the form (4.5), we have at t=O
u(x,O)= F(x) + G(x) = f(x),
ut(x, O) = CF'(X) - cG'(x) = g(x).
(4.9)
(4.10)
u(x,O)=f(x),
Differentiating (4.9) with respect to x and solving the two linear equations
for F' and G we obtain
I
F'(x)
Cf'(X)+g(x)
2c
or,
F(x)=f~) +
G'(x)= cj'(x)- g(x)
2c
;c foxg(~)d~+8,
(4.11)
(4.12)
G(x)=f(x) __
I rxg(~)d~+e,
2
2c)0
with suitable constants 8, e. Here 8 + e = 0 by (4.9). Hence
u(x,t) = F(x+ ct) + G(x- ct)
I
=-2I (J(x + ct) + f(x - ct)) +-2
c
f
x + ct
x-ct
g(~)d~.
(4.13)
For f E C 2 and gEe 1 this actually represents a solution u E C 2 of the
initial-value problem (4.1), (4.8).
We see from (4.13) that u(x,t) is determined uniquely by the values of
the initial functionsJ,g in the interval (x, -ct,x+ct) of the x-axis whose
end points are cut out by the characteristics through the point (x,t). This
interval represents the domain of dependence for the solution at the point
(x, t) as shown in Figure 2.2.
---r----------~--------------------~--------x
o
x - ct
x
+ ct
Figure 2.2
39
4 The One-Dimensional Wave Equation
--~------------~----~--------~-------------x
o
x -ct
x
+ ct
Figure 2.3
Conversely, the initial values at a point (~,O) of the x-axis influence
u(x,t) at points (x,t) in the wedge-shaped region bounded by the characteristics through (~, 0), i.e., for ~ - ct < x < ~ + ct. This indicates that for
our equation "disturbances" or "signals" only travel with speed c as shown
in Figure 2.3.
We saw that formula (4.5) represents a solution uEC 2(1Ji 2) of (4.1) for
any j,g, E C 2(1Ji). One is tempted to consider any u of the form (4.5) for
"general" j,g, as a generalized or weak solution of (4.1) even though u may
not have derivatives in the ordinary sense. One easily verifies that any
function u of the form (4.5) satisfies the functional equation
u(x,t) - u(x + cr,t+ r) - u(x- CT/,t+ 11)
+ u(x+ Cr-CT/,I+ r+11)=O. (4.14)
Geometrically, for any parallelogram ABCD in the xl-plane bounded by
four characteristic lines, (see Figure 2.4), the sums of the values of u in
opposite vertices are equal, that is,
u(A)+u(C)=u(B)+ u(D).
(4.15)
Every solution of (4.1) is of the form (4.5) with F, G, E C 2 and thus satisfies
(4.14). Conversely, using Taylor expansions for small r,11, every C 2-solution of (4.14) satisfies (4.1). Thus (4.14) can be viewed as a weak formulation of equation (4.1).
We use (4.4) to solve an "initial-boundary-value" problem for (4.1).
Assume the wave equation to be satisfied only in a fixed x-interval
0< x < L for all I> O. Then we can prescribe, in addition to the initial data
u=j(x),
40
ut=g(x)
forO<x<L, t=O
(4.16)
2 Second-order equations
A
D
B
c
--~------------------------------------x
o
Figure 2.4
certain "boundary" conditions, for example
u=a(t) forx=O
O<t,
u=/3(t) forx=L
O<t.
We are interested in the solution of (4.1) in the strip
(4.17a)
(4.17b)
(4.18)
O<x<L,
O<t.
We divide the strip into a number of regions by the characteristics through
the comers and through the points of intersections of the characteristics
with the boundaries, etc. as shown in Figure 2.5.
In region I the solution u is determined by the formula (4.13) from the
initial data alone. In a point A = (x, t) of region II we form the characteristic parallelogram with vertices A, B, C, D and get u(A) from (4.14) as
u(A)=-u(C)+u(B)+u(D),
(4.19)
with u(B) known from boundary condition (4.17a) and u(C),u(D) known
since C,D lie in I. Similarly, we get u successively in all points of the
regions III, IV, V, .... If we want the solution u of this "mixed" problem to
be regular (e.g., to be of class C 2 ) in the closure of the strip, the data
f,g, a, /3 have to fit together in the comers so that u and its first and second
derivatives come out to be the same when computed either from f,g or
from a,/3. We clearly need the compatibility conditions,
a (0) = f(O),
a'(O) = g(O),
a"(O) = c2j"(O)
/3(0)= f(L),
/3'(0) = g(L),
/3"(0) = c2j"(L).
(4.20)
41
2 Second-order equations
u =a(x)
o
u
u
= f(x),
Ut
= (3(x)
x
= g(x)
Figure 2.5
These actually are also sufficient for uEC 2 whenj,a,/3 EC 2 and gEC l •
For example, for A EI and u(A) given by (4.19) we take the limit as A~D
for D fixed. Then u(B)~a(O), u(C)~j(O) and u(A)~- j(O) + u(D)+a(O)
= u(D), if (4.20) holds. If, instead, a(O):;6f(O), we would find that u has a
jump all along the line x = ct.
An alternative method ("separation of variables") gives the solution at
one stroke by expansion into eigenfunctions. For simplicity, take the case
where L = 'IT and a = /3 = 0 in (4.11), so that we have homogeneous boundary
conditions. Then, for each t, u(x,t) can be expanded into a Fourier sine
series
00
u = ~ an (t)sinnx.
(4.21)
n=l
Substituting into (4.1) we find that ait) satisfies the ordinary differential
equation + n 2c 2an = 0, hence that an is of the form
a::
an(t) = Cn cos(nct) + dnsin(nct).
(4.22)
Here the constants cn,dn can be found from the initial conditions (4.16)
which require that
00
j(x)= ~ cnsinnx,
n-l
42
00
g(x) = c ~ ndnsinnx,
n-l
(4.23)
4 The One-Dimensional Wave Equation
so that by Fourier
en=~ ('lTf(x)sinnxdx,
dn =2en'fT
'fT )0
i'ITg(x)sinnxdx.
0
(4.24)
In applications we usually deal only with a bounded domain for our
solutions, and are led to initial-boundary-value problems rather than to
"pure" initial-value problems. An example is the normal displacement
u(x,l) of a vibrating string of length 'fT. Then u satisfies the boundary
conditions u(x,O)=u(x,'fT)=O, if the ends of the string are held fixed, and
initial conditions of type (4.16) if the initial normal' displacement and
normal velocity of each particle of the string are prescribed.
PROBLEMS
1. Let j(x),g(x) have compact support (i.e., vanish for all sufficiently large Ixl).
Show that the solution u(x,t) of (4.1), (4.8) has compact support in x for each
fixed t. Show that the functions F, G in the decomposition (4.5) for U can be of
compact support only when f~oog(€)~=O.
2. Let a be a constant =1= -c. Find the solution u(x,t) of (4.1) in the quadrant
x>O,t>O, for which
u=j(x),
ut=g(x) for t=O, x>O
ut=aux for x=O, t>O,
wherejandg are of class C 2 for x>O and vanish near x=O. (Hint: Use (4.5».
Show that generally no solution exists when a = - c.
3. Solve the initial-boundary-value problem ("mixed problem")
Utt=Uxx forO<x<'1T, 0< t,
u=O for x=O,'1T; 0< t,
u=l,
ut=O forO<x<'1T,t=O,
by (a) piecing together, using (4.14), and (b) Fourier series. Check that the
solutions agree.
4. Let the operators
LI'~
be defined by
L1u=aux+buy+cu,
~u=dux+~+ ju,
where a,h,c,d,e,j are constants with ac- hd=l=O. Prove that
(a) the equations L1u= WI> ~u= W2 have a common solution u, if L1W2= ~~I'
(Hint: By linear transformation reduce to the case a = e = I, h = d = 0.)
(b) The general solution of LI~u=O has the form U=UI+u2, where L1UI=0,
L 2U2=0.
5. Solve
Utt - c2uxx = x 2 for 0 < t
u=x,
and all x
ut=O for t=O.
(Hint: First find a special time independent solution of the P.D.E.)
6. Find a solution of
43
2 Second-order equations
(A constant) of the form u=f(x 2 -c2t1=f(s), when f(O) = I, in form of a
power series in s.
7. With L defined by (4.1) prove that
(a) Lu=O, Lv=O implies L(utvt + cluxvx ) =0,
(b) Prove thatfor Lu=O, Lv=O for a< x<b, 1>0 and u=O for x=a,b; 1>0.
d (bl
dl J~ 2 (utVt + cluxvx)dx =0.
a
8. For the solution (4.21), (4.22) of the wave equation express the "energy"
('IT I
Jo
2
2
2(ut +ux)dx,
in terms of cn' dn•
9. Find the Fourier series solution (4.21) for the case
u(x,O)=i-li-xj.
Ur(x,O)=O forO<x<'IT,
(vibration of string plucked at center) and calculate its energy.
10. Find in closed form the solution u(x, I) of
Lu=utt-cluxx=O for O<X, 0< I,
u(O,/)=h(/) for 0<1,
u(x,O)=f(x)
ut(x,O)=g(x) forO<x,
withf,g,h E C 2 for nonnegative arguments and satisfying
h(O)=f(O),
h'(O) = g(O),
h"(O) = c2j"(O).
Verify that the u obtained has continuous second derivatives even on the
characteristic line x = cl.
5. Systems of First-order Equations
It is convenient to treat general second-order equations as part of a still
more general theory, that of first-order systems, to which in principle all
higher-order single equations can be reduced. Thus the linear equation
(3.1) for u(x,y) is reduced to a first-order system by introducing the new
dependent variables
(5.1)
yielding the equations
(5.2)
(5.3)
44
5 Systems of First-order Equations (Courant-Lax Theory)
More generally, writing t for y, we consider an N-vector (column vector)
u{}U(X,t),
satisfying an equation
au
A(x,t)ai+B(x,t)
au
ax = C(x,t)u+D(x,t),
(S.4)
(S.5)
for given square matrices A,B,C of order N and column vector D. The
Cauchy problem for (5.5) prescribes the value of u on a curve t=<t>{x) in
the xt-plane
(S.6)
u= u(x,cp(x» = j(x).
The curve is characteristic if we cannot find the derivatives of u from the
data on the curve. Now (5.5), (5.6) imply on t=<t>{x) that
(S.7)
Aut+Bux = Cj+D,
or
(5.8)
(A -cp' B )ut = Cj+ Dd- BJ'.
Thus, characteristic curves are those for which the matrix A - cp' B is
singular, or for which the Nth-degree differential equation
det(Adx-Bdt)=O
(S.9)
holds.
For the initial-value problem to be treated here we prescribe the values
of u on the x-axis, assumed to be non-characteristic:
(S.lO)
u(x,O)= j(x),
(S.11)
detA *0 for t=O.
For small t we can then solve for Ut in (5.5) getting a new equation
ut+B(x,t)ux=Cu+D
(S.12)
(With new matrices B, C and vector D.) The characteristic differential
equation (5.9) now takes the form
det(
~~ / -
B ) =0
(5.13).
(/ = unit matrix) which factors into conditions of the form
dx
dt =Ak(X,t),
(S.14)
where ~(x,t) denotes the kth eigenvalue of the matrix B(x,t). We assume
these eigenvalues to be real, so that for each k (S.14) is satisfied by a
one-parameter family of characteristic curves Ck •
More precisely, we assume the system (S.5) to be hyperbolic in the sense
45
2 Second-order equations
that there exists a complete set of real eigenvectors ~ 1, ••• , ~N of B such that
Be = Ake,
(5.15)
where the ~k are linearly independent and depend "regularly" on x and t.
(This certainly holds where the eigenvalues of B are real and distinct.) The
column vectors
then form the columns of a nondegenerate matrix
f=f(x,t) for which
e
(5.16)
Bf=fA,
where A is the diagonal matrix whose diagonal elements are the eigenvalues Ak •
Introducing a new unknown vector v by u = fv, we find from (5.12) that
v satisfies
(5.17)
with new coefficients
c=f-1Cf-f-1ft -f-1Bfx'
and initial conditions
for t=O.
(5.18)
We have reduced (5.5) to "canonical form" where A =1 and B is
diagonal. If v has components Vi' c = (Cik ), and d has components di we find
from (5.17), along a characteristic Ci of the ith family,
v=f-Y=g(x)
dVi =
dt
aVi + dx aVi = aVi +\ aVi = L C-kVk+d..
at dt ax at ax k
I
(5.19)
I
The ith backward characteristic Ci through a point (X, T) has an equation
(5.20)
(obtained by solving the ordinary differential equation (5.14)). Then by
(5.18), (5.19), integrating along Ci (see Figure 2.6)
vi(X, T) = gi (ai (0, X, T»
+
[T( ~
CikVk + d; )dt,
(5.21)
where in the integrand x has to be replaced by a;(t, X, T). Formula (5.21)
resembles a system of integral equations except that the domain of integration is different for each component of v. We write (5.21) symbolically as
v= W+Sv,
(5.22)
where W is the vector (considered known) with components,
W;(X, T) = gi(ai(O,X, T»
+loTdi(ai(t,X, T),t)dt
(5.23)
and S is the linear operator taking a vector v with components vk(x,t) into
46
5 Systems of First-order Equations (Courant-Lax Theory)
(X, T)
__
~~
____L -__- L______
~
______
~
______L -_____ x
Figure 2.6
a vector w = Sv with components
w;(X, T) =
r ~ C;k( a;(t,X, T),t)Vk( a;(t,X, T),i)dt.
T
)0
k
(5.24)
Given sufficient regularity of our data, the mapping S: C~C is continuous
in the space C of continuous bounded vectors v(x,t) with domain in the
strip 0 ~ t ~ T, using in C the "maximum norm":
Ilvll =
sup
k=l, ... ,N
IVk(X,t)l·
(5.25)
x;O<I<'T
The norm IISII of the operator S (the supremum of IISvll for Ilvll = 1)
obviously is bounded * by the constant
q=T sup ~ IC;k(X,t)l.
i,x
k
O<I<'T
(5.26)
For q < 1, i.e., for T sufficiently small, the mapping S is contractive. Then
(5.22) has a unique solution vEe obtainable by the process of iteration
(5.27)
(Convergence follows by comparison with a geometric series.)
The resulting v will satisfy the "integral equations" (5.21), but it is not
certain that these imply the differential equations (5.19) or (5.17) until we
have established existence of continuous partial derivatives for v. For that
purpose we shall work in the narrower Banach space C 1 of vectors vex, t)
for which v and Vx are continuous and bounded for 0 ~ t ~ T and all x, in
which we choose as norm
(5.28)
IlIvlll =max(lIvll, Ilvxll)·
·We assume, for convenience, that the C;k(X,t) and their first derivatives are bounded
uniformly for - 00 < x < 00, 0.;; t.;; T.
47
2 Second-order equations
The restriction of S to C 1 maps C 1 into itself and has its norm for C 1
bounded by
q*= '1"(
~ (!~~ lcilcx(x,/)1 + Icjk(x,/)I) t~~;)ajx(/,X, T)I)
+'1" ~ SUpICjk(X,/)I.
k
(5.29)
x,t
For q* < I equation (5.22) has a solution in C I, provided WE C I. Obviously Sv for v E C ~ can be differentiated with respect to T as well. Thus,
convergence of the v n and v: implies convergence of vtn • Hence in the limit
we obtain a solution v of the original initial-value problem for our partial
differential equations. The condition q* < I will again be satisfied for '1"
sufficiently small. In this way, we obtain an existence theorem for a
sufficiently small I-interval. Here v can be continued as long as our
ordinary differential equations for the characteristics lead to sufficiently
regular solutions permitting us to define functions a;(/,X, T) describing
characteristics from (X, T) backwards to the x-axis. Further extensions to
quasi-linear systems were given by Courant and Lax.
We have solved the pure initial-value problem in the linear case,
assuming that the initial data are known on the whole x-axis. The method
described extends to mixed initial-boundary-value problems as well. For
simplicity take as domain for the vector v the first quadrant x > 0, I> O.
We then need, in addition to initial conditions,
v=g(x) for O<x, 1=0,
(5.30)
boundary conditions on the half line x = 0, 0 < I. The number of these
conditions depends on the number of positive eigenvalues Ak of the matrix
B. Let, say, the number r be such that
(5.31)
The characteristics Cj of the ith family then have positive slopes for
i=I, ... ,r, negative ones for i=r+I, ... ,N. A backward characteristic
X=a;(/,X,T)
(5.32)
through a point (X, T) with 0 < X, 0 < T for i = r + I, ... , N will hit the
positive x-axis in a point Pj(X,T). For i=I, ... ,r, it will either hit the
positive X-axis or positive t-axis in a point, again denoted by P;(X, T). On
the positive I-axis we prescribe the values of r linear combinations of the
components of v, which we assume to have been brought into the form
n
Vj-
~
k=r+1
Yjk(t)Vk=hj(t)
for 0< I, x=O;i= 1, ... ,r.
(5.33)
We redefine the operator S acting on vectors v. Let, for a vector v(x,/), the
component Wj(X, T) of W= Sv be defined by (5.24) if Pj(X, T) lies on the
48
5 Systems of First-order Equations (Courant-Lax Theory)
x-axis, in particular, whenever i=r+ 1•... ,N. If, on the other hand,
P;(X, T) lies on the t-axis, say P; = (0, to), we take
N
N
w;(X, T)= ~ 'Y;k(tO) Lto ~ cks(ak(t,O,to),t)v.. (ak(t,O,to),t)dt
k-r+l
N
+~
k= 1
0 s=r+l
fto c;k(a;(t,X,T),t)vk(a;(t,X,T),t)dt.
T
(5.34)
(Here the second sum corresponds to the expression for v;(X, T) in terms
of v;(P;) and integrals along C;, the first sum corresponds to v;(P;) expressed by the Vk(P;) for k> r, which, in turn, are expressed by integrals
over backward characteristics through P; [See Figure 2.7].) We again get a
set of equations formally describable by (5.22) with a known W, which can
be solved for 0 <:; t < 'T, with 'T sufficiently small. Solutions of problems in a
finite interval 0 <:; x <:; L can be reduced to pure initial-value problems, and
to problems of the type just discussed, by breaking up the domain
0", x <:; L, 0 < t <:; 'T into suitable portions by characteristics, as was done
earlier for the one-dimensional wave equation. The number of boundary
conditions on x = 0 equals the number of positive Ak , that on x = L equals
the number of negative Ak •
(X. T)
Figure 2.7
49
2 Second-order equations
PROBLEM
Write
IItt=c'luxx ,
u(x,O)=f(x),
ut(x,O)=g(x)
as an initial-value problem for the vector (Ul,U2) = (IIt,Ux )' Reduce the system to the
canonical form (5.17) and solve the problem.
6. A Quasi-linear System and Simple Waves
Consider a quasi-linear system of equations for functions v1(x, t), ... ,
vN(x,y) forming a column vector v(x,t), of the special form
vt+B(v)vx=O
(6.1)
with a square matrix B(v) depending on the dependent variables. We
assume that (6.1) is hyperbolic in the sense that B(v) has real distinct
eigenvalues for all v in question. The general solution v(x, t) can be
visualized as forming a two-dimensional surface in N-dimensional v-space,
referred to the two parameters x,t. Special explicit solutions can be
obtained by requiring that this surface degenerates into a curve, that is,
that the range of the solution v is one-dimensional. Such v can be
represented in the form
(6.2)
v=F(O),
where the scalar 0 is a function of x and t. Substituting into (6.1) yields
F'(O)Ot+B(F)F'(O)Ox=O.
(6.3)
Thus F'(O) must be an eigenvector of the matrix B(F) belonging to an
eigenvalue A such that Ot + AOx = O. These conditions can be satisfied by
thinking of B(v) as constituting a field of matrices in lV-dimensional
v-space. With each point v we associate an eigenvector V = V( v) and
corresponding eigenvalue A=A(V), varying smoothly with v. We take a
vector F= F(O) correspond.ng to a particular solution of the system of
ordinary differential equations
dF
(6.4)
dO = V(F).
Along this solution the eigenvalue A(V)=A(F(O» becomes a known function c(O). Taking for O=O(x,t) any solution of the scalar equation
°t+c(O)Ox=O
(6.5)
we have in v=F(O(x,t» a solution of (6.1) called a simple wave.
The solution of (6.5) with initial values
O=</>(x) for t=O
is given by the implicit equation [See Chapter I, (6.28)].
o=</>(x- c(O )t).
50
(6.6)
6 A Quasi-linear System and Simple Waves
PROBLEM
Solve
u//-(1 + Ux )2Uxx =0
with initial conditions (containing an arbitrary function h)
u=h(x),
u,= -h'(x)-th,2(x) for t=O.
(Hint: Convert to a system for the vector v with components
[Answer:
VI
= Ilx, V2 = u/.)
u=h(x-(1 +9 )t)+Ih,2(x-(1 +9 )t),
with 9(x,t) given implicitly by
9=h'(x-(1 +9)t).]
Find the domain of existence of u when h(X)=X2.
51
3
Characteristic manifolds
and the Cauchy problem*
1. Notation of Laurent Schwartzt
This multi-index notation is extremely convenient for partial differential
equations, keeping us from drowning in a flood of subscripts. We consider
here "vectors" ~ = (~]' ... ,~) with n (usually) real components and "multiindices" a=(a], ... ,a,,) which are vectors whose components are nonnegative integers. With a multi-index a we associate the scalars
a! =a]!a2!' .. a,,!.
lal=a] +a2+··· +an ,
From a vector
~
(Ll)
and multi-index a we form the monomial
~a = ~1.~22
... ~n"".
(1.2)
By Ca we generally denote a coefficient depending on n nonnegative
integers a], •.. ,a,,:
Ca = Ca •... ..".
The general mth-degree polynomial in n variables
form
P(~) =
~
lal<m
Ca~a.
(1.3)
~], ... '~n
is then of the
(1.4)
Using the Cauchy differentiation symbol Dk = a/aXk' we introduce the
"gradient vector" D =(D], ... ,Dn ), and define the gradient of a function
u( X], ••• , xn) as the vector
(1.5)
*([7D
t([12], [15D
52
1 Notation of Laurent Schwartz
The general partial differentiation operator of order m is then
am
(1.6)
where /a/=m.
For a function f( x) = f(x!, ... , x n ) we have the formal power series
expansion
f(x+y)=
L J,(D'1(x))ya.
a.
a
(1.7)
We callf(x) real analytic in a region 0 of Rn if for each xEO the series on
the right of (1.7) converges absolutely and represents f( x +y) for all
sufficiently small vectors y. We call f( x) real analytic at a point XO if f( x) is
defined and real analytic in a neighborhood of XO in Rn. It is clear that a
functionf(x) which is real analytic at the origin of Rn can be extended by
its power series to complex x as a function with continuous derivatives of
all orders in a neighborhood of the origin in complex n-space en. Conversely let f(x) be defined and continuous in a complex neighborhood of
the origin and let the first derivatives of f exist and be continuous in that
complex neighborhood. Thenf(x) is real analytic in a neighborhood of the
origin of Rn. Indeed by repeating the ordinary Cauchy formula for
functions of one complex variable, we derive for f near 0 in Rn the multiple
integral representation
where the paths of integration are circles of sufficiently small radius about
the origin in the complex plane. Expanding the integrands one sees
immediately that f(x) is represented by a convergent power series for all
sufficiently small /xk /. The most conspicuous property of real analytic
functionsf(x) is that of unique continuation: Iffis real analytic in an open
connected set 0 C Rn, then f is already determined uniquely everywhere in
o by its values in an arbitrarily small neighborhood of some point of O.
PROBLEMS
Let x,y denote a vector and a,{3 multi-indices with n components. Prove
the following identities:
1. the binomial theorem
(1.9)
53
3 Characteristic manifolds and the Cauchy problem
2. Leibnitz's rule for scalar functionsf(x),g(x)
L
Da(Jg)=
f>
p,y
a~!, (DPj)(DYg)
f'
.y.
(1.10)
f3+y='a
3. geometric series: for IXkl < 1 for k= l, ... ,n
~ x a = -=-_--.,---l--;-::--~
(l-x\) ... (l-xn )
a
(1.11 )
4. for a nonnegative integer m
(1.12)
(1.13)
2. The Cauchy Problem*
In the Schwartz notation the general mth-order linear differential equation
for a function u(x)=u(x\, ... ,xn) takes the simple form
Lu=
L
lal<m
Aa(x)D"u= B(x).
(2.1)
The same formula describes the general mth-order system of N differential
equations in N unknowns if we interpret u and B as column vectors with N
components and the Aa as N X N square matrices. Similarly the general
mth-order quasi-linear equation (respectively system of such equations) is
Lu=
L
lal=m
AaDau+ C=O,
(2.2)
where now the A and C are functions of the independent variables Xk and
of the derivatives D f3u of the unknown u of orders IPI..; m - 1. More
general nonlinear equations or systems
F(x,Dau) =0
(2.3)
can be reduced formally to quasi-linear ones by applying a first-order
differential operator to (2.3). On the other hand, an mth-order quasi-linear
system (2.2) can be reduced to a (larger) first-order one, by introducing all
derivatives Df3u with IPI..; m-l as new dependent variables, and making
use of suitable compatibility conditions for the Df3u.
The Cauchy Problem consists of finding a solution u of (2.2) or (2.1)
having prescribed Cauchy data on a hyper-surface S C IRn given by
l/>(x\, ... ,xn)=O.
* ([23D
54
(2.4)
2 The Cauchy Problem
Here cp shall have m continuous derivatives and the surface should be
regular in the sense that
(2.5)
The Cauchy data on S for an mth-order equation consist of the derivatives
of u of orders less than or equal to m - 1. They cannot be given arbitrarily
but have to satisfy the compatibility conditions valid on S for all functions
regular near S (instead normal derivatives of order less than m can be
given independently from each other). We are to find a solution u near S
which has these Cauchy data on S. We call S noncharacteristic if we can
get all Dfiu for lal = m on S from the linear algebraic system of equations
consisting of the compatibility conditions for the data and the partial
differential equation (2.2) taken on S. We call S characteristic if at each
point x of S the surface S is not noncharacteristic. Characteristic surfaces
naturally occur in connection with singular solutions of a certain type. If u
is a (generalized) solution of (2.1) of class C m - l which has jump discontinuities in its mth-order derivatives along a surface S, then S must be a
characteristic surface. (See Chapter 2, p.34.)
To get an algebraic criterion for characteristic surfaces we first consider
the special case where the hyper-surface S is the coordinate plane Xn = O.
The Cauchy data then consist of the Dfiu with 1,81 < m taken for t=O.
Singling out the "normal" derivatives on S of orders .s; : m - 1:
D:U=o/k(X1,,,,,Xn_ l )
for k=O, ... ,m-l and xn=O
(2.6)
we have on S
DfiU=Df'Df2 ... D:~-Nfi"
(2.7)
provided that ,8n < m. In particular for I,81.s;;: m - 1 we have here the
compatibility conditions expressing all Cauchy data in terms of normal
derivatives on S. Let a* denote the multi-index
a*=(O, ... ,O,m).
(2.8)
In the differential equation (2.1) or (2.2) taken on S it is only the term with
%, ... ,o/m-l and hence in"
terms of the Cauchy data. All others contain derivatives D"u with lanl.s;;:
m - 1. Thus Da*u, and hence all D"u with lal.s;;: m, are determined uniquely
on S, if we can solve the differential equation for the term D a*u. This is
always possible in a unique way if and only if the matrix A a * is nondegenerate, i.e., det(Aa*)~O. For a single scalar differential equation this condition reduces to Aa*~O. In the linear case the validity of the condition
a = a* that is not expressible by (2.7) in terms of
(2.9)
does not depend on the Cauchy data on S; in the quasi-linear case,
however, where the Aa depend on the Dfiu with I,81.s;;: m-l and on x, one
has to know the o/k in order to decide if S is noncharacteristic.
55
3 Characteristic manifolds and the Cauchy problem
Condition (2.9) involves coefficients of mth-order derivatives. We define
the principal part Lpr of L (both in (2.2) and (2.1» as consisting of the
highest order terms of L:
(2.10)
Lpr= ~ AQDQ.
IQI-m
The "symbol" of this differential operator is the matrix form ("characteristic matrix" of L):
A(O= ~ AQr a •
(2.11)
lal=m
Here the N x N matrix A(n has elements that are mth-degree forms in the
components of the vector r = (r \, ... ,rn). In particular, the multiplier of D;
in Lpr is Aa*=A("1), where
"1=(O, ... ,O,I)=Dcp.
(2.12)
is the unit normal to the surface cp = xn = 0. The condition for the plane
cp = xn = to be noncharacteristic is then
Q(Dcp) *0,
(2.13)
where Q = Q(n is the characteristic form defined by
(2.14)
Q(r)= det(A(r»
for any vector r. (In the case of a scalar equation (N = 1) the characteristic
form Q(n coincides with the polynomial A(r).) We shall see that quite
generally (2.13) is the condition for a surface cp=O to be noncharacteristic.
Take now a general S described by (2.4). By assumption (2.5) the first
derivatives of cp do not vanish simultaneously. Suppose that in a neighborhood of a given point of S, the condition CPxn *0 holds. The transformation
°
y.=
I
{
Xi
cp(x\, ... ,xn )
fori=I, ... ,n-1
for i= n
(2.15)
is then locally regular and invertible. By the chain rule,
au ""
aXi
au
= ~ Cik aYk '
(2.16)
where the
(2.17)
are functions of X or of y. Denoting by C the matrix of the Cik and
introducing the gradient operator d with respect to y with components
a
di=-a '
~i
(2.18)
we can write (2.16) symbolically, as
D=Cd,
56
(2.19)
2 The Cauchy Problem
taking D and d to be column vectors. Generally, then for
lal = m
Da={Cdr+Ra,
(2.20)
where Ra is a linear differential operator involving only derivatives of
orders "m - 1, (arising from the dependence of C on x) and (Cd)a is
formed as if C were a constant matrix, i.e., not applying differentiations to
the elements of C. Then the principal part of the operator L in (2.2) or
(2.1) transformed to y-coordinates is given by
Lpr= ~ AaCCdr=tpr
lal=m
(2.21)
and its symbol, the characteristic matrix of t, by
A{7J)= ~ Aa{C7Jr·
lal=m
(2.22)
For the regular mapping (2.15) x-derivatives of orders less than or equal to
r are linear combinations of y-derivatives of orders less than or equal to r,
and conversely. Hence noncharacteristic behavior of S is preserved under
the transformation. Thus S is noncharacteristic for L if the plane y n = 0 is
noncharacteristic with respect to the operator L transformed to y-coordinates, i.e., if
det{A{7J))=det(
~
lal=m
AaCC'l/r)=Fo,
(2.23)
for the column vector 7J with components (0, ... ,0, 1). But then S= C7J is
just the column vector with components aYn/ih; = D;q" that is, the vector
Dq,. Thus, the condition for noncharacteristic behavior of S can again be
written as (2.13).
If u in (2.2) stands for a vector with N components, the condition for S
to be a characteristic surface
Q{Dq,)=det(
~
lal=m
Aa{Dq,r)=o
(2.24)
signifies the vanishing on S of a form of degree Nm in the components of
Dq,. In the linear case (where the coefficients of that form only depend on
x and not on u) we can consider a one-parameter family q,(Xl""'Xn)=
const. = c of characteristic surfaces. Then (2.24) becomes a first-order
partial differential equation of q" homogeneous of degree Nm in the first
derivatives of q, from which q, can be determined by the methods of
solution for single first-order equations. For example in the case of a linear
first-order system
n
Lu= ~ A;{x)
a
a~; +B{x)u=w{x),
(2.25)
(u and w being N-vectors and the A; and B, N X N matrices) the condition
57
3 Characteristic manifolds and the Cauchy problem
for a characteristic surface is
det(
~ ~<t> Ai) =0.
i= I uXi
(2.26)
Alternately a single characteristic surface S gives rise to a partial differential equation, when described by an explicit equation
<t>(x l , ... ,Xn) = Xn - tf;(x l ,. ",Xn-I) =0.
(2.27)
Equation (2.24) in the linear case becomes a first-order partial differential
equation for the function tf;. Take, e.g., the wave equation
(2.28a)
for u= U(XI,X2,t). A characteristic surface t=tf;(xI,x0 then satisfies the
equation
1 = c2( tf;;. + tf;;,)
(2.28b)
already encountered in Chapter 1, p. 26. Thus equation (2.28b) regulates
the propagation of singularities for equation (2.28a).
The characteristic form Q(n of the operator L defined by (2.14), (2.11)
generally depends on and on the arguments of the Aa, that is, on x and
the DPu with IPI..;; m - 1. A hyperSurface is characteristic for the operator
L (and in the nonlinear case for given Cauchy data Dfiu) if Q(n=O for the
normal vector r of the surface. We call L elliptic if Q(n,,=O for all real
r,,= o. In that case there exist no real characteristic hypersurfaces. In the
case of an operator L with real coefficients, ellipticity of L is then
equivalent to definiteness of the form Q(r) (at least for n> 1): the form
Q(n is of constant sign for r,,=O. This, of course, can only happen when
the degree mN of Q is even. The standard example for an elliptic L is the
+ ... + for which
Laplace operator !J. =
r
Dr
D;
n
Q(r)= ~ r/
i= I
is positive definite. The extension of the notion of "hyperbolicity" to
general quasi-linear systems is more complicated.
We can define characteristic manifolds for more general nonlinear
equations as well. Take, for example, an mth-order scalar equation for a
function u=u(xl, ... ,xn), which we write in the form
(2.29a)
where Pa = Dau for lui..;; m. Suppose that on the hypersurface S given by
(2.4) we have <t>x. ,,=0. Differentiating (2.29a) with respect to Xn we obtain
the equation
dF "'"
dF
0=1)+
~ 1) Dn Dau .
r
Xn
a
'Pa
(2.29b)
Since rn"=O for = D<t>, the condition for S to be characteristic with respect
58
3 Cauchy-Kowalewski Theorem
to (2.29b) is simply
~ ~F (D4>t =0.
I~m 'Pa
(2.30)
We use (2.30) to define what is meant by S to be characteristic with respect
to (2.29a). Observe that this condition generally involves not only the
Cauchy data but all derivatives of u of orders ~ m on S. In the example of
the first-order equation (7.1) of Chapter 1, condition (2.30) for a characteristic (projection) 4>( x,y) = const. would read
F;,4>x + Fq4>y = O.
Since here 4>x dx + cf>y dy = 0 along S, the condition is equivalent to the
second equation in (7.10) of Chapter 1.
PROBLEMS
1. Identify the special cases of "characteristic curves" in Chapter 2 (1.12) and
(5.13) with the general formula (2.13).
2. Let u(X)=U(Xb""Xn ) and its derivatives of orders <m vanish on the hypersurface S given by (2.4). Show that on S
D"u=JL(DCPr for JaJ=m,
(2.31)
where the factor of proportionality JL depends on u but not on a. Show that in
particular for u of the form u=cpmv(x) with vEC m
D"u=m!(DCPrv.
(Hint: Transform S by (2.15) into the planeYn=O.)
(2.32)
3. Cauchy-Kowalewski Theorem
This theorem concerns the existence of a solution of the Cauchy problem
formulated earlier for the case of analytic data and analytic equations. We
restrict ourselves to quasi-linear systems of type (2.1), since more general
nonlinear systems can be reduced to quasi-linear ones by differentiation.
We assume that the initial surface S is analytic in a neighborhood of one
of its points xo, that is near Xo the surface S is given by an equation
4>(x)=O, where 4> is real analytic at xo, (that is, given by a convergent
power series in x - xo) and that D4>=I=O at xo, say, Dn4>=I=O. On S we
prescribe compatible Cauchy data DPu for 1,8l ~ m - 1 which shall be real
analytic at x-xo (e.g., represented by power series in (XI-X?, ... ,Xn_lx~
The coefficients Aa and B shall be real analytic functions of their
arguments x and DPu at xo, that is, given by convergent power series in
x-xo and in (DPu-DPu~ in a neighborhood of xo, where DPuo is the
value of DP corresponding to the Cauchy data at xO. Let, moreover, S be
noncharacteristic at Xo (and hence in a neighborhood of x~ in the sense
that Q(D4»=I=O. Then the Cauchy-Kowalewski theorem asserts that there
-I»'
59
3 Characteristic manifolds and the Cauchy problem
exists a solution of the Cauchy problem which is analytic at XO (given by a
power series in x - xC) and that there is no other analytic solution.
The proof of this general theorem consists of showing that all
coefficients for a prospective power series solution u at XO can be obtained
by successive differentiation from the differential equation and Cauchy
data, and that the resulting series actually converges to a solution.
The proof becomes easier if we reduce the problem before constructing
the power series. First of all, one transforms S locally by an analytic
transformation into a neighborhood of the origin in the plane Xn = O. Then
by introducing derivatives of orders less than or equal to m - I as new
dependent variables one reduces the system to one of the first order. We
make use of the fact here that the set of real analytic functions is closed
under differentiation and composition. One arrives at a first-order system
in which the coefficient matrix of the term with aujaxn is nondegenerate
because S is noncharacteristic. Hence one can solve for aujdxn, obtaining
a system in the standard form
n-I
au
~
~= ~
j=1
uXn
au
Aj(x,u)""il+ B(x,u),
uXj
(3.1)
where the Aj(x,u) are square matrices (aijk) and B(x,u) a column vector
with components bj' analytic in their arguments. Written out componentwise, (3.1) becomes a system for the N components t4.i of u:
n-I
N
L aijk(x,u)DjUk+ bj(x,u).
i= 1 k= 1
Dnt4.i= ~
(3.2)
On xn=O, near 0, we have prescribed initial values u=f(x). Here we can
assume that f = 0, introducing u - f as the new unknown function. Thus,
is the initial condition for (3.1) or (3.2). By assumption the aijk and bj are
given by convergent power series in the xp Ur near x = u = O.
Once all derivatives of u(x) at x=O have been determined, u, if analytic,
is determined uniquely and given by the Taylor series. Now this determination is carried out easily from (3.2) and (3.3). Indeed (3.2) permits
us to express successively all derivatives Dau in terms of derivatives DfJu
with f3n=O. First, by (3.2), Dnt4.i is expressed in terms of x,u,Dju with i=t=n;
then also DYDnu, for Yn =0, by applying DYu to (3.2). Next, differentiating
(3.2) with respect to x n' we find D;u in terms of x, u,
Dju,Dnu,DjDkU,DjDnu, with i,k=t=n, and thus in terms of x,u,Dju,DjDku
and then also DYDn2u for Yn=O. Continuing in this way we find an
expression for any D~ in terms of x,u,DfJu with f3n=O. More precisely,
this expression will be a polynomial in the DfJu and in derivatives of the
aijk(x,u) and /J.i(x,u) with respect to the Xj and Ur with coefficients which
are nonnegative integers. Putting then x=O, all DfJu with f3n=O vanish and
60
3 Cauchy-Kowalewski Theorem
we find that at the origin,
D~m = Pa,m(DY8paijk,DY8pbj)'
(3.4)
where DY indicates differentiation with respect to x and 813 with respect to
u, all derivatives taken at the origin, and Pa,m is a polynomial with
nonnegative coefficients. One sees this more clearly from the model of the
ordinary differential equation
u'=b(x,u)
for a scalar function u(x) of the scalar variable x with initial condition
u(O) = O. One finds successively for x = 0,
u=O,
u'=b,
etc.
Clearly an analytic solution u(x), if it exists, is given by
uj =
L l,(D~j)xa,
a.
a
(3.5)
with Dauj obtained from (3.4). It remains to show that the formal series
converges for Ixl sufficiently small and that it actually represents a solution
of (3.2), (3.3). Convergence in (3.5) depends on getting appropriate estimates for the expression (3.4) for the D~j' This is difficult because of the
complicated structure of the polynomials and can be avoided by just using
the positive character of the coefficients. The arguments of the PaJ are the
derivatives of the aijk and bj with respect to the X k and Ur at x = u =0. These
in tum differ from the coefficients of the power series for the aijk and bj in
terms of x and u only by positive factors. Let us replace the differential
equation (3.2) by
(3.6)'
with initial condition
v=O for xn=O
(3.6a)
and assume (3.6), (3.6a) has a solution v analytic at O. Then the coefficients
of its power series are determined by
Davm= Pa,m( D Y8 Paijk,D Y8 Pbj).
(3.7)
Assume now that aijk' bj majorise aijk' bj in the sense that
I( D Y8 Paijk) x=u=ol ,,( D Y8 Paijk) x=u=o'
I( D Y13 Pbj )x= u=ol ,,( D Y8 Pbj) x=u=o'
for all f3 and y. Then in (3.4), (3.7)
Ipa,m( Dr8paijk,Dr13Pbj)I" Pa,m( D r13 paijk,D r8 Pbj)
(3.8)
61
3 Characteristic manifolds and the Cauchy problem
(with derivatives taken at x = u = 0). Hence, since the series
~ ~! (Davj\=ox a
(3.9)
converges absolutely in a neighborhood of the origin, the same holds for
the series (3.6), by comparison.
It is thus sufficient to find a majorising equation (3.6) for which
existence of an analytic solution v satisfying the initial condition (3.6a) can
be established. Consider generally a function F(x)=F(xl, ... ,xn) real
analytic at x = O. Let
(3.10)
F(x) = ~ Caxa
a
for IXkl...;; r, k= 1, ... ,n. (The series will also converge for complex Xk with
IXkl < r, but we shall not make use of this fact). Since the series converges
for XI = ... = Xn = r we have Carla l bounded:
ICal...;; Mr-Ial...;; la~! Mr- Ial .
(3.11)
a.
for a certain M. The function
G(x)= ~ C~xa,
(3.12)
a
analytic at 0, majorises F(x) if
(3.13)
We write then
F«G.
In particular, by (Lll), (1.13), (3.11),
F«~ Mr-Ialxa=
a
Mr n
(r-x l )'" (r-xn)
«~ lal! Mr-Ialxa=
a a!
(3.14)
(3.15)
Mr
r-xl-x2-"'-xn
Here r measures the size of the complex neighborhood IXkl <r, k=
1,2, ... ,n, of 0 in which F(x) is analytic, and M controls the magnitude of
F in that complex neighborhood.
Thus there exist suitable constants M,r, such that
b(
)
.~u«
J
62
M(n-l)r2
(r - x I - ... - xn-I - u I
-'"N
- u )(r
- x )
n
.
(3.16)
3 Cauchy-Kowalewski Theorem
Hence we can majorise problem (3.2), (3.3) for u by one of the form
D
Mr2
V.=~-------------:-;-----,­
nJ
(r-x,-'"
where tJ =
°for
-xn_,-v,-'" -vN)(r-xn )
Xn
= 0. The solution of this problem has the form
V,=V2='"
=vN =
1
N(w(s,t)-s),
(3.18)
where
(3.19)
Here the scalar w(s, t) is simply the solution of a Cauchy problem for a
single first-order equation
w=
t
Mr2(n-l)
w
(r-w)(r-t) s'
(3.20a)
(3.20b)
w(s,O)=s.
This problem can be solved explicitly by the methods of Chapter 1 and one
verifies easily that its solution is analytic at the origin.
One still has to verify that the analytic vector u constructed from (3.4),
(3.5) actually satisfies (3.1). We omit the somewhat tedious but not difficult
argument showing this. This then completes the proof of the CauchyKowalewski theorem.
PROBLEMS
1. Solve (3.20a,b) and find a positive p such that w(s,t) is analytic for complex s,t
with lsi < p, Itl < p, and hence such that the power series for w(s, t) converges for
those s,t. (Answer:
w=~(r+s-vr-si+4M(n-l)r210g( I-f) ).
Possibly p=r/(2+32M(n-I).)
2. In the linear case the system (3.2) can be given the form
n-'
Dn~= ~
N
~ aijk(x)DjUk+ ~ bjk(x)Uk+Cj(X),
i=' k='
k='
(3.21)
It can be majorised by a system of the form
Dn Vj = (r-s )~ r-t ) (M~
i,k DjVk + M~
k Vk + IL)
(3.22)
63
3 Characteristic manifolds and the Cauchy problem
with s,t defined as in (3.19) and with suitable constants r,M,p.. Solve (3.22) with
for t-O. Show that the resulting region in which the solution u of (3.21)
can be represented by a power series depends on r and M but not on p. (and
hence not on the "size" of the Cj in a complex neighborhood of the origin).
Vj-O
3. Let the scalar function F(x) of the scalar variable x be defined for
power series
00
Ixl < I by the
k
F(x)=c ~ \ '
(::ik
where C is a positive constant
(a) Express F in integral form.
(b) Prove F2«F for C sufficiently small.
4. Observe that the initial-value problem for the scalar ordinary differential equation
u'=I+u2 withu(O)=O
is majorised by the problem
1
v'=-- with V (0) =0.
I-v
Hence deduce an upper bound for the power series expansion of u(x)=tanx.
4. The Lagrange-Green Identity
We recall the Gauss divergence theorem:
where d / dn denotes differentiation in the direction of the exterior unit
normal
n ) of ag and dx-dxl ... dxn , dSx-surface element with
of our region to be
integration on x. We always assume the boundary
sufficiently regular so that the divergence theorem applies to all u E C 1(0).
The theorem can be generalized to- u E C 1(g) n CO(O) by approximating g
from the interior. More generally, we have the formula for integration by
parts,
r-(rl .... ,r
an
(4.2)
where u, v are column vectors belonging to C I(g) with T denoting transposition.
Let now L be a linear differential operator
Lu- ~ aa(x)D"u.
lal<m
(4.3)
Let u,v be column vectors and aa be square matrices in Cm(O). Then by
64
5 The Uniqueness Theorem of Holmgren
repeated application of (4.3) it follows that
L
g
vT
~ aa(x)D~dx
lal<m
=
J:o
~ (-I)la IDa(v Ta,,(x»)udx+ r M(v,u,ndSx' (4.4)
J ao
lal<m
Here M in the surface integral is linear in the !k with coefficients which are
bilinear in the derivatives of v and u, the total number of differentiations
in each term being at most m - 1. The expression M is not determined
uniquely but depends on the order of performing the integration by parts.
This is the Lagrange-Green identity for L which we also write in the form
r vTLudx= r (iv)TUdx+ r M(v,u,ndSx,
J
J
(4.5)
iv= ~ (- I)1"IDa(aa(x)Tv).
lal<m
(4.6)
Jg
where
i
o
ao
is the (formally) adjoint operator to L, defined by
The simplest example corresponds to the Laplace operator L = tl for
scalars u and v. Then one integration by parts yields
Jrovtludx= Jrao ~i vUx,!;dS- Jro ~i vrux,dx.
-,
(4.7)
We write this as
Jao vddudS-r~vxuxdx.
n
Jo i "
Ivtludx= (
o
(4.8)
Integrating once more by parts we obtain
1'0vtludx= kutlvdx+
IaJ ~~
v
- u ~~ )dSx'
(4.9)
5. The Uniqueness Theorem of Holmgren
It is clear from the arguments used in the proof of the Cauchy-Kowalewski theorem that an analytic Cauchy problem with data prescribed on an
analytic noncharacteristic surface S has at most one analytic solution u,
since the coefficients of the power series for u are determined uniquely.
This does not exclude the possibility that other nonanalytic solutions of the
same problem might exist. However, uniqueness can be proved for the
Cauchy problem for a linear equation with analytic coefficients and for
data (not necessarily analytic) prescribed on an analytic noncharacteristic
surface S. The method of proof (due to Holmgren) makes use of the
Cauchy-Kowalewski theorem and the Lagrange-Green identity. (Extension of the uniqueness theorem to nonanalytic equations is much more
difficult).
65
3 Characteristic manifolds and the Cauchy problem
Figure 3.1
We give a sketch of the Holmgren argument which reduces uniqueness
of solutions U(XI""'Xn ) for one problem to existence of solutions of the
"adjoint" problem for a dense set of data. Assume we are given an analytic
family of hypersurfaces SA depending on a parameter A. for 0..; A...; 1 and
such that all SA have a common (n - 2)-dimensional boundary. For 0..; A. <
/L"; 1, the surfaces SA'S,. shall fonn the boundary of an n-dimensional
region RAw (See Figure 3.1.) All S>.. shall be noncharacteristic with respect
to the mth-order linear operator
L= ~ Aa(x)Da,
(5.1)
lal<m
where the Aa are analytic in RO]. Holmgren's theorem asserts that a
solution of Lu = 0 with Cauchy data 0 on So is detennined uniquely in R OI •
Assume we know already that u has vanishing Cauchy data on S>... If there
exists for a certain w(x) a solution v of
iv=w(x)
(5.2)
in R>..p' for which
on S,. for
IPI < m, then
(5.3)
f
wTudx=O
(5.4)
RA,~
by the Lagrange-Green identity (4.5). If here the w for which a solution v
of (5.2), (5.3) exists fonn a dense set* of functions in CO(R>..p) it follows that
• The set is dense in the sense that any continuous u can be approximated uniformly by w in
the set.
66
6 Distribution Solutions
u=O in RAp. and hence also that u has vanishing Cauchy data on S"..
Actually the Cauchy-Kowalewski theorem only asserts that for given
analytic w there exists a solution v of (5.2) and (5.3) in a neighborhood of Sp.
and not necessarily in all of RAp.. However, a closer look (not carried out
here; see problem 2 at the end of Section 3) at the proof permits us to show
that in the linear case a solution of (5.2) and (5.3) will exist in all of RAp. for
all polynomials w provided only that I /L - AI is sufficiently small. Since the
polynomials w form a complete set by the Weierstrass approximation
theorem, it follows that u = 0 in R}..p.. Dividing now the interval 0« A« I
into a finite number of sufficiently small subintervals by points AO=O,
Al, ... ,AN= I, we prove successively that u=O in R}..O}..I,R}..I}..Z,R""_I}..N' and
hence in all of R01 •
6. Distribution Solutions*
The Lagrange-Green identity (4.5) has other applications. The identity
reduces to
k
vLu dx =
k
(iv) T u dx
(6.1)
in the case when all boundary terms vanish, e.g., when either u or v have
Cauchy data zero. In particular, let
Lu=w inn and DfJv=O onanforIPI<m.
Then the identity
(6.2)
(6.3)
holds. We can use this identity to define generalized solutions u of Lu=w.
For example, if u is continuous we can require this identity to hold for all
vE em, vanishing near an. This leads to the notion oj distribution solutions
in the sense of Laurent Schwartz.
The idea is to replace a functionJ(x) which is defined on an open set n
in ~n by the integrals formed with this function for different weights w. We
associate with the point function J( x) the Junctional
f[ <p] = k<P(x)J(x)dx,
(6.4)
where <P E 6j) = CoOO(n) is the space of "test functions," that is, functions
having derivatives of all orders and compact support. t This Junctional J
exists for any continuous or locally integrable Junction J. The integral for
J[<p] defines a linear functional on 6j), with values in ~. The values of the
functionJ[<p] for varying <P determine the functionJ(x) uniquely whenJ(x)
is continuous. Indeed, if the continuous function g is such that J[<P] = g[<p]
* ([3], [8], [15])
tTbat is, cj>=O outside a closed and bounded subset of D. Generally, the support of a function
cj>(x) is the closure of the set of x for which cj>(x),.,O.
67
3 Characteristic manifolds and the Cauchy problem
for all cP E 6j), then,
f cp(x)(j(x) - g(x))dx=O
(6.5)
for all CPE6j). If here f¢g, say f- g>O at a point P, then also in a
neighborhood of P; choosing a test function which is nonnegative and
vanishes outside this neighborhood would contradict (6.5). Thus f- g=O.
Consider next derivatives of f. If f E C l(n) we find by integration by
parts
(6.6)
The left-hand side is obtained by applying the junctional D,j to test
functions cpo Thus D,J[cp] = - fIDkCP]. We can use the right-hand side of
(6.6) to define the left-hand side when f has no derivative or is not even
continuous. As long as the functional associated with f is defined for all
test functions cp, the functional associated with D,j makes sense. More
generally, we are led to the notion of a distribution.
Definition. A distribution is a linear functionalfIcp] defined for all CPE6j) =
CoOO(n) which is continuous on 6j) in the following sense: Let the CPr be a
sequence in 6j). Then
provided
(a) all CPk vanish outside the same compact subset of
(b)
n, and
uniformly in x for each a (not necessarily uniformly in a).
Each continuous (or even locally integrable) function f(x) generates a
distribution
f[ cp ] =
f cp(x)f(x)dx.
(6.7)
More generally, we write any distribution f symbolically as
k
f[ cpJ = cp(x)f(x)dx.
(6.8)
A specially important distribution, not generated by an integrable point
function, is the so-called Dirac function with singularity ~, denoted by 8~,
which is defined by
(6.9)
It is symbolically given by
10cp(x )8~(x)dx =cpW·
68
(6.10)
6 Distribution Solutions
Two distributions f[cp] and g[cp] naturally are called equal, if f[cp] = g[cp]
for all cp E 6j). More generally we say that two distributions f,g agree in an
open subset w of 0 if f[cp] = g[cp] for all cp E 6j) that have their support in w;
This permits us in some cases to assign point values to a distribution in a
subset of O. Thus for the Dirac function 81< defined by (6.9) we have
8E(x)=0 for xEO, x*~.
(6.11)
Indeed takingf= 8E, g=O, we havef[cp]= g[cp] =0 for any cp with support in
the set w obtained by deleting ~ from O.
For a distribution f we define the "derivative" D,J as the distribution
given by
(6.12)
and more generally D af by
D"l[ cp ] = ( - I)lalJ[ Dacp ].
(6.13)
One easily verifies that formulas (6.12), (6.13) actually define distributions
D,J,D&.j. As an example we have from (6.9) that Dk 8l<[cp] = -CPXk(O. In
particular, (6.13) yields a definition for derivatives of a continuous function f(x), not necessarily as a function with point values, but as a
generalized function for which weighted integrals are defined.
Still more generally than (6.13) we can apply any scalar linear differential operator L with C<XJ coefficients to a distribution u(cp), we define the
distribution Lu(cp) by
Lu[ cp] =u[ £p]
(6.14)
in accordance with the Lagrange-Green integral formula
~cpLudx= ~{£P)udx
(6.15)
valid by (4.5) for a test function </> and a scalar function uEC m • (Similar
definitions can easily be given for the case where u is a vector, and L
corresponds to a system.) We are interested particularly in the distribution
solutions L of the equation
(6.16)
where 81< is the Dirac operator defined by (6.9). They are the so-called
fundamental solutions with pole ~ for the operator L. We notice that adding
to a fundamental solution u any ordinary solution v E Cm(O) of the
homogeneous equation Lv = 0 again yields a fundamental solution.
In many cases the equations defining a particular distribution as a linear
operator in 6j) define the operator for a much wider variety of functions cp.
For example, the operator 8E[cp] can be defined by (6.9) or (6.10) for all
functions cp(x) that are just continuous in O. Similarly for an mth-order
linear differential operator L with coefficients in Cm(O), we can define
Lu(cp) by (6.14) for a locally integrable function u(x) and for cpE Cm(O).
More precisely we call u( x) a weak solution of the equation Lu( x) = w( x) in
69
3 Characteristic manifolds and the Cauchy problem
n, if
f <J>(x)w(x)dx=f (i<J>(x»u(x)dx
(6.17)
for all <J> E C~(n) (that is for all <J> of class c m with compact support in n)*.
Other types of generalized solutions will be encountered in the sequence.
EXAMPLE. Let n be the x-axis and L the ordinary differential operator
d 2 jdx 2 • Then u(x)=~lx-~1 is a fundamental solution for L. Indeed the
distribution associated with u" by (6.13) is
L:oo ~lx-~I<J>"(x)dx=<J>(~)= 8€[
u"[ <J> ] = u[ <J>"J =
when
<J> E
<J> ]
COOO(IR).
PROBLEMS
1. Show that for a continuous function f the expression u = f(x - ct) is a weak:
solution of the partial differential equation
ut+cux=O.
(Hint: Transform for <p E CJ(1R2) the integral
f f (<pt+c<Px)udxdt
to the coordinates YI = X
-
ct, Y2 = x.)
2. Show that the function U(X\>X2) defined by
U (X
I,
X
)= { I
2
0
for XI >~I' x2>~2
for all other X\>X2
is a fundamental solution with the pole (~I' ~2) of the operator L = a2/ ax I aX2 in
the xlxrplane.
3. Show that the function
u ( XI,X2 ) =
{
!
2
o
forlxl-~d<~2-x2
otherwise
is a fundamental solution for L=(a2/ax~)-(a2/ax?) with pole (~1'~2)'
4. Verify the special cases of Green's identity
(6.18)
dv
du
d/).v) dS.
v--/).u-+/).v--uLo(v/).u-u/).v)dx= iao (d/)'U
dn
dn
dn
dn
2
2
* It is sufficient to require (6.17) to hold for all </> E 6j) since every
mated uniformly with its derivatives of order .;; m by </> E 6j) •
70
</> E
(6.19)
C{)' can be approxi-
6 Distribution Solutions
5. Show from Holmgren's uniqueness theorem that a solution u(X,y,z,/) of the
wave equation Ult = c2( Uxx + ~y + uzz ) at the point (0,0,0, T) is determined
uniquely by its Cauchy data on
x2+y2+z2<c 2T2,
1=0.
(Hint: Take for the S suitable hyperboloids of revolution about the I-axis
passing through X2+y2+z2=c 2T2, 1=0.)
71
4
The Laplace equation *
1. Green's Identity, Fundamental Solutions,
and Poisson's Equationt
The Laplace operator acting on a function u( x) = u( X I' ••• ,xn ) of class C 2 in
a region Q is defined by
n
d= ~
k=!
Df
(1.1)
For u,v E C 2 (Q) we have:!: (see Chapter 3, (4.8), (4.9» Green's identities.
where d / dn indicates differentiation in the direction of the exterior normal
to ClQ.
The special case v = I yields the identity
r dudx= JrafJ ddun dS.
JfJ
(1.3)
*([2]. [6]. [11]. [13]. [14]. [14]. [23]. [26D
t({17D
*We assume here that
Chapter 3. is valid.
72
n is an open bounded set, for which the divergence theorem, (4.1) of
I Green's Identity, Fundamental Solutions, and Poisson's EqUation
Another special case of interest is v=u. We find then from (1.2a) the
energy identity
r~ u~dx+ JrouAudx= Jrao u un
~u dS.
Jo
i
(1.4)
If here Au = 0 in Q and either u = 0 or du / dn = 0 on aQ, it follows that
r ~i u;dx=O.
'
Jo
(1.5)
For u E C 2(n) the integrand is nonnegative and continuous, and hence has
to vanish. Thus u = const. in Q. This observation leads to uniqueness
theorems for two of the standard problems of potential theory:
The Dirichlet problem: Find u in Q from prescribed values of Au in Q
and of u on aQ.
The Neumann problem: Find u in Q from prescribed values of Au in Q
and of du / dn on aQ.
As always in discussing uniqueness of linear problems, we form the
difference of two solutions, which is a solution of the same problem with
data O. We find that the difference is a constant, which, in the Dirichlet
case, must have the value O. Thus: A solution u E c 2(n) of the Dirichlet
problem is determined uniquely. A solution u E C 2(n) of the Neumann
problem is determined uniquely within an additive constant. (Notice also that
the solution of the Neumann problem can only exist if the data satisfy
condition (1.3».
One of the principal features of the Laplace equation
Au=O
(1.6)
is its spherical symmetry. The equation is preserved under rotations about a
that is under orthogonal linear substitutions for x-~. This makes
it plausible that there exist special solutions v(x) of (1.6) that are invariant
under rotations about ~, that is have the same value at all points x at the
same distance from ~. Such solutions would be of the form
point~,
v=l[;(r),
(1.7)
where
(1.8)
represents the euclidean distance between x and ~. By the chain rule of
differentiation we find from (1.6) in n dimensions that I[; satisfies the
ordinary differential equation
n-l
Av= I[;"(r) +-I[;'(r) =0.
r
(1.9)
73
4 The Laplace equation
Solving we are led to
(1.10a)
I//(r) = Cr 1 - n
Cr2-n
I/;(r)= { 2-n
Clogr
whenn>2
(1.10b)
whenn=2
(1.1Oc)
with C = const., where we can still add a trivial constant solution to 1/;.
The function v(x)=I/;(r) satisfies (1.6) for r>O, that is for x+~, but
becomes infinite for x=~. We shall see that v for a suitable choice of the
constant C, is a fundamental solution for the operator d, satisfying the
symbolic equation, (see Chapter 3, (6.16)),
(1.11)
Let u E C 2(Q) and ~ be a point of n. We apply Green's identity (1.2b) with
v given by (1.7), (1.10b,c). Since v is singular at x=~ we cut out from 12 a
ball B(~,p) contained in 12 with center ~, radius p, and boundary S(~,p).
The remaining region np = 12 - B(t p) is bounded by an and S(~, p). Since
dv = 0 in np we have
r vdudx= JrafJ (v dudn -u dn.
dv )dS+ r
J
JfJ
S(g,p)
p
(v du -u dV)dS.
dn
dn
np
Here on S(~,p) the "exterior" normal to our region
Consequently by (1.10a,b,c), (1.3)
points towards
v = I/;(p),
J.r
S(g,p)
du
vTdS=I/;(p)
n
r
f
~.
(1.13a)
S(g,p)
Tdu dS = -I/;(p)
n
u dv dS= - Cpl-n
JS(g,p) dn
(1.12)
r
JS(g,p)
f
B(g,p)
udS.
dudx
(1.13b)
(1.13c)
Since both u and du are continuous at ~ the right-hand side of (1.13b)
tends to 0 for p~O by (1.10b,c), while that of (1.l3c) tends to - CWnu(~),
where Wn denotes the surface "area" of the unit sphere in IRn. [The values
W2-2'IT, w3=4'IT are familiar; generally Wn =2Y7T n/r(~n)]. Thus (1.12)
becomes for p~O
(note that v, though
74
00
at
x=~,
is integrable near
~.
We now choose
I Green's Identity, Fundamental Solutions, and Poisson's Equation
C= ljwn in (1.10b,c), so that
1/I(r)={
(2::;W
logr
for
(USa)
n>2
n
for n=2.
2'11"
We write the corresponding v in its dependence on x and
(USb)
~
as
v = K(x,~) = 1/I(r) = 1/1(1 x -~I).
(1.16)
Then (1.14) becomes
u(~)= Jr K(x,~)Audx- Jr (K(X'~) d~(x)
-u(x) dKd,X'~) )dSx (1.17)
un
nx
o
ao
x
for ~E~, where the subscript "x" in Sx and dnx indicates the variable of
integration respectively differentiation. (Notice that for ~ e: D the left-hand
side of identity (1.17) has to be replaced by O. This follows from (1.2b),
since v=K(x,~EC2(D) and Av=O for ~ outside D.)
Taking in particular for u in (1.17) a test function </> E COoo(~), we find
that
f
</>(~)= K(x,~)A</>(x)dx
Hence v =
K(x,~
(1.18)
defines a distribution for which
v[ A</> ] = </>(~).
Since L=A is (formally) selfadjoint (that is L=L), we can interpret (1.17)
as stating that the functional vA applied to a test function </> has the value
</>(~) (see Chapter 3, (6.15», or that v in the distribution sense satisfies
(1.11), and is a fundamental solution with pole ~.
Let uE C 2(D) be "harmonic" in~, that is, be a solution of Au=O; then
by (1.17) for ~E~
u(~)= -
r (K(X'~) dUdnx(x) -u(x) dKd,X'~)
)dSx.
nx
J ao
(1.19)
Formula (1.19) expressing u in ~ in terms of its Cauchy data u and dr!i dnx
on a~ represents the solution of the Cauchy problem for ~, provided such
a solution exists. Actually, by the uniqueness theorem for the Dirichlet
problem proved earlier a solution of Au = 0 is determined already by the
values of u alone on a~. Thus we cannot prescribe* both u and du j dnx on
a~. The Cauchy problem Jor the Laplace equation in ~ generally has no
solution. Formula (1.18), however, is useful in discussing regularity of
harmonic functions. Since K(x,~)=1/I(lx-~D is in Coo in x and ~ for X=F~,
we can form derivatives of u with respect to
~
of all orders under the
• The Cauchy data cannot even be prescribed as arbitrary ana(ytic functions on aD. This does
not contradict the Cauchy-Kowalewski existence theorem, since here we require u to exist in
all of D, and not just in a sufficiently small neighborhood of aD.
75
4 The Laplace equation
integral sign for ~ E~, and find that u E C oo(~). More precisely, we can
even conclude that u(~ is real analytic in ~. To see this we only have to
continue u suitably into a complex neighborhood. We observe that K =
t/!(r), with r given by the algebraic expression
is defined and differentiable for complex x,~ as long as r+O. In particular
r+O when x is confined to real points on im and ~ to complex points in a
sufficiently small complex neighborhood of a real point of ~. Formula
(1.19) then defines u(O as a differentiable function of ~ in that complex
neighborhood. It follows that u is real analytic in the set ~. We need not
assume that u E C 2(U). To belong to Coo or to be analytic is a "local"
property. A solution of the Laplace equation in any open set ~ is of class
C 2 in any closed ball contained in ~ and hence real analytic in the open
ball. Thus harmonic functions are real analytic in the interior of their domain
of definition.
We can show from this that the Cauchy problem for the Laplace
equation generally is unsolvable, even locally. Take for the initial surface a
portion (1 of the plane xn = O. Prescribe the Cauchy data
u=O,
(1.20)
on (1. Let ~E (1 and let B be a ball with center ~ whose intersection with
Xn = 0 lies in (1. Denote by B + the hemispherical portion of B lying in
xn ;) O. There cannot exist a solution u of the Laplace equation of class C 2
in jj + with Cauchy data (1.20) for xn = 0, unless g is real analytic. To see
this one continues u into the whole of B by reflection, defining u(x)=
u(x1, ... ,xn) for xEB with xn<O by
(1.21)
One easily verifies that the extended u belongs to C 2(B); (the values of u
and its first and second derivatives fit the reflected values along xn = 0,
because u=~u=O there). Moreover, ~u=O in B, since the Laplace equation is unchanged when we replace u by - u or Xn by - Xn. Consequently u
is real analytic in the whole of B. In particular then ux,,(x1, ... ,Xn_I,0) is a
real analytic function of XI"'" Xn - I'
Let w(x)=w(xl, ... ,xn) be any solution of ~w=O of class C 2(U). Then
G(X,~) = K(x,~) + w(x)
(1.22)
again is a fundamental solution of the Laplace equation with pole ~. (See
Chapter 3, p. 69.) More precisely identity (1.17) stays valid when we
76
I Green's Identity, Fundamental Solutions, and Poisson's Equation
replace K by G:
1
u(g) =
n
G(x,g)tludx-
i
m
(G(X,g)
d~(x) ~
as follows immediately from (1.17), (1.2b).
As an application we take the case where
and radius p, choosing for G the function
Then on
u(x) dGd;,g) )dSA1.23)
x
n is a ball B(g,p) of center g
an
dG _.//(p)_
- -1p I-n ,
dnx
Wn
G=O,
--'t'
and (1.23) becomes
u(g)=
I
Ix-gl<p
(1[;(lx-gj}-1[;(p»tlu(x)dx+
In_I
WnP
I
Ix-fl=p
u(x)dSx •
(1.2S)
For tlu=O this is Gauss's law of the arithmetic mean:
u(g)
In_I
WnP
I
Ix-fl=p
u(x)dSx '
(1.26)
where wnpn-I is the surface area of the sphere Ix-gl=p. In words: For a
function u harmonic in a closed ball the value of u at the center equals the
average of the values of u on the surface. Since the 1[;(r) given by (1.1Sa,b) is
a monotone increasing function of r for all dimensions n, we obtain more
generally from (1.25):
If tlu(x) ~ 0 in the ball Ix -gl .,;; p, then
u(g).,;;
In_I
WnP
I
Ix-gl=p
u(s)dSx '
(1.27)
A function u cont41uous in n is called subharmonic, if for each gEn the
inequality (1.27) holds for all sufficiently small p. Thus functions in C 2(n)
with tlu";; 0 are subharmonic.
Formula (1.18) expresses that K(x,g) is a fundamental solution for tl.
Another aspect of this property is Poisson's formula
(1.28)
vaiid for u E C 2(U) and gEn. Here tl€ denotes the Laplace operator taken
with respect to the variables g. Thus for given u E C 2(U) we have in
w(g) = ~ K(x, g)u(x) dx
(1.29)
77
4 The Laplace equation
a special solution w of the inhomogeneous Laplace equation ("Poisson's
differential equation")
(1.30)
~~w(~)=u(~)
for
~EQ.
Formally (1.28) follows directly from (1.16), (1.11):
~~f K(x,~)u(x)dx= f (~~lHlx-W)u(x)dx
= f (~A(lx-~I»u(x)dx
= f 8~(x)u(x)dx= u(~).
For a rigorous derivation we first assume that u E CJ(Q). Then by (1.17)
for ~ E Q, since u = 0 near aQ,
L
u(~)= K(x'~)~xu(x)dx= f K(x'~)~xu(x)dx.
(1.31)
(As always the domain of integration is the whole space, when no other
domain is indicated). Actually (1.31) holds for all ~, since for given
u E CJ(Q) nothing changes when Q is replaced by any larger open set
containing~. Withy=x-~ as variable of integration we have by (1.16)
u(~)= f l{I(lx-W~xu(x)dx= f l{I(lyi)~u(y+~)t6-'
=f
l{I(IYI)~~u(y +~)t6-' =~~f l{I(lyi)u(y +~)t6-'
=~~f K(x,~)u(x)dx
confirming (1.28). Let next u E C 2(Q). Let b be any ball with ii c Q. We can
find a concentric ball B such that ii c B, jj c Q, and a "cutoff function"
nx)ECJ(Q), which has the value 1 everywhere in B. (See Figure 4.1.)
Then
where
ru
u=ru+(I-nu,
E CJ(Q) and (1vanishes in B. Thus for ~ E b
nu
~"L K(x,~)Hx)u(x)dx= rwu(~)= u(~)
~~ r K(x,~)(l- Hx»u(x)dx =~" r
J~
J~-B
(1.32a)
K(x,~)(I - r(x»u(x)dx=O,
(1.32b)
since K(x,~) is regular and harmonic in ~ for xEQ-B, ~Eb. Adding
(1.32a,b) yields (1.28) for ~ in any b, and hence for all ~EQ.
For ~ outside Q we find by direct differentiation under the integral sign
that ~~w(~)=O for the w defined by (1.29). For ~ on aQ the second
78
1 Green's Identity, Fundamental Solutions, and Poisson's Equation
~=O
I
/
/
/
/
.".-
I
\
\
"'"
-
'""'----
----'
Figure 4.1
derivatives of w({l may cease to exist. One easily verifies however that
C l(lRn), since the first ~-derivatives of K( x,~) are still integrable with
respect to x.
w(~) E
PROBLEMS
1. Let u belong to C 2(Q) and be subharmonic. Show that au ;;.0. [Hint: Use
(1.25).]
2. Let L=a+c in n=3 dimensions, where c=const. (L="reduced wave opera-
tor").
(a) Find all solutions of Lu=O with spherical symmetry.
(b) Prove that
K(x,~)= _ cos~Vc r) ,
wr
(1.33)
is a fundamental solution for L with pole ~. Show that for a solution u of
Lu=O formula (1.19) holds with K defined by (1.33). [Hint: Use formula
(6.18) of Chapter 3.]
(c) Show that a solution u of Lu = 0 in the ball Ix - ~I ;;;; p has the modified mean
value property
u(~)=
1
Vc p ~
u(x)dSx •
sin(Vc p) 4wp Ix-EI=p
(1.34)
(Here for c = - y < 0, the factor Vc p/sin(Vc p) stands for
Vy p/sinh(Vy p).) [Hint: Use the fundamental solution
(1.35)
with a suitable constant k.]
79
4 The Laplace equation
(d) Show that a solution u of Lu=O of class C 2(U) vanishing on ao vanishes in
0, provided c < O. Show that for c > 0 there are solutions vanishing on a
sphere but not in the interior.
(e) Show that solutions of Lu=O in 0 are real analytic in O.
3. (a) Show that for n=2 the function
v=_1 r 210gr
r=lx-~1
(1.36)
8'IT
'
is a fundamental solution for the operator /)/. [Hint: dv=(1 + logr)/2'IT.]
(b) Show that for u E C 2(U) and ~ E 0
u(~)=lvd2udx-r (vddU_dUdv+dVdu_uddV)dS.
o
Jao dn
dn
dn
dn
(1.37)
[Hint: Apply (1.2b) with u replaced by dU, and use (1.17).]
4. (a) Find all solutions with spherical symmetry of the biharmonic equation
d2u=o in n dimensions.
(b) Find a fundamental solution.
S. Prove that u(x)=u(xJ> ... ,xn)=O also implies that
(1.38)
d(lxI2- nU(x/lxI2») =0
for x/lxl2 in the domain of definition of u.
6. Let x,y denote coordinates in the plane. Let u(x,y) be a solution of the
two-dimensional Laplace equation Uxx + Uyy = 0 in an open simply connected set
O.
(a) Prove that there exists a conjugate harmonic function v(x,y) such that the
Cauchy-Riemann equations
ux=Vy,
(1.39)
Uy= -Vx
are satisfied. [Hint: for a fixed point (xo,Yo) in 0 define v by
(1.40)
where the integral is taken along any path joining (xo,Yo) to (x,y).]
(b) Introduce the complex-valued function f = u + iv of the complex argument
z = x + ry. Prove Cauchy's theorem that for any closed curve C in 0
I
f(z) dz =
Ie
(u + iv)( dx + idy) = O.
[Hint: There exist functions q,(x,y), 1jJ(x,y) in 0 with
dq,=udx-vdy,
d!V=vdx+udy.]
7. By Newton's law the gravitational attraction exerted on a unit mass located at
~= (~1>~2'~3) by a solid 0 with density JL = JL(x) is given by the vector
F(~)=y
Iff JL(X)(X-~)
o
(y = universal gravitational constant).
80
Ix-~13
dx
2 The Maximum Principle
(a) Prove that F= gradu, where the "potential" u is given by
u(~)=Yfff
p.(x) dx.
Ix-~I
(b) Prove that the attraction F(f> exerted by 0 on a far away unit mass is
approximately the same as if the total mass of 0 were concentrated at its
center of gravity
xO=
f fDf p.(x)xdx/ f fDf p.(x)dx.
[Hint: Approximate 1~-xr3 by 1~_xol-3 for large I~I.]
(c) Calculate the potential u and attraction F of a solid sphere 0 of radius a
with center at the origin and of constant density p.. Use here that u must
have spherical symmetry, must be harmonic outside 0, satisfy Poisson's
differential equation in 0, be of class C 1 everywhere, and vanish at 00.
[Answer: u(~)=2'ITyp.a2-~'lT'Yp.1~12 for 1~I<a,u(f>=~'lT'Yp.a31~1-1 for 1~I>a.]
8. Let u, v be conjugate harmonic functions satisfying (1.39) in a simply connected
region O. Show that on the boundary curve ao
du dv
dv
du
dn=ds'
dn=-ds'
where dn denotes differentiation in the direction of the exterior normal and ds
differentiation in the counterclockwise tangential direction. Show how these
relations can be used to reduce the Neumann problem for u to the Dirichlet
problem for v and conversely.
2. The Maximum Principle
One of the important tools in the theory of harmonic functions is the
maximum principle. Similar principles hold for solutions of more general
second-order elliptic equations, and for complex analytic functions. In this
section we assume that ~ is a bounded, open, and connected set in Rn. We
first prove a weak Jorm oj the principle:
Let u E C 2(O) n CO(Q), and let flu ;;. 0 in O. Then
maxu=maxu.
fi
aD
(2.1)
(Notice that a continuous function u assumes its maximum somewhere in
the closed and bounded set D. Formula (2.1) asserts that u assumes its
maximum certainly on the boundary of ~, possibly also in ~.)
Indeed under the stronger assumption flu>O in~, relation (2.1) follows
from the fact that u cannot assume its maximum at any point ~E~; for
then a2u / ax~ '" 0 at ~ for all k, and hence flu '" O. In the case flu ;> 0 in ~
we make use of the auxiliary function v=lxl 2 for which flv>O in~. Then
for any constant e > 0 the function u + w belongs to C2(~) n CoCO), and
satisfies fl(u+w»O in~; hence
max(u+w)= max(u+w).
Li
aD
81
4 The Laplace equation
Then also
maxu+eminv< maxu+emaxv.
n
n
an
an
For e~O we obtain (2.1).
In the special case where u is harmonic in g, relation (2.1) also applies
to - u. Since minu = - max( - u) we obtain
min u = min u.
n
(2.2)
an
These relations imply a maximum principle for the absolute value (using
that lal =max(a, - a) for real a):
If u E C 2(g) n Co(U) and ~u = 0 in g, then
maxlul=
max lui.
n
an
(2.3)
In particular u = 0 in g if u = 0 on ago This implies an improved uniqueness
theorem for the Dirichlet problem (see p. 73) not requiring u to have
derivatives on the boundary:
A function u E C 2(g) n Co(U) is determined uniquely by the values of
~u in g and ofu on ago
Stricter versions of the maximum principle flow from the mean value
theorem (1.27):
Let u E C 2(g) and ~u ~ 0 in g. Then either u is a constant or
u(~) <sup u
n
for all ~Eg.
(2.4)
More generally (2.4) holds for any nonconstant u that is subharmonic in g.
For the proof we set M=supu, and decompose g into the two disjoint
sets
The set g2 is open because of the continuity of u. Using the fact that u is
subharmonic we can show that gl is open as well. For if ~Egl' we have
from (1.27) for all sufficiently small p
0<1
Ix-~I=p
=
I
u(x)dSx-Wnpn-lu(~)
Ix-~I=p
(u(x)-u(~)dSx=
I
Ix-~I=p
(u(x)-M)dSx·
Since u( x) - M is continuous and <0, it follows that u( x) - M = 0 on every
sufficiently small sphere with center ~. Hence all x in a neighborhood of ~
82
2 The Maximum Principle
belong to ~l' and ~l is open. By definition a connected open set cannot be
decomposed into two disjoint open nonempty sets. Thus either ~l or ~2 is
empty, proving the principle.
As an immediate consequence we have:
IfuEC2(~)n CO(U) and ilu;;;'O in~, then either u=const. or
u(~) <max u
an
(2.5)
for all ~ E~.
Again (2.5) holds more generally for all u that are continuous in U and
subharmonic in ~. (We need only observe that max u cannot be assumed
at a point of ~.)
PROBLEMS
1. Let Q denote the unbounded set JxJ > 1. Let uE C 2(n), Llu=O in Q, and
limx->cou(x)=O. Show that
max JuJ = max u.
aD
~
[Hint: Apply maximum principle to spherical shell.]
2. Let u( x) E C 2(Q) n C 0(0) be a solution of
n
(2.6)
Llu+ ~ ak(x)ux +c(x)u=O,
k=l
k
where c(x) <0 in Q. Show that u=o on aQ implies u=O in Q. [Hint: Show that
maxu.;; 0, minu ~ 0.]
3. Prove the weak maximum principle (2.1) for solutions of the two-dimensional
elliptic equation
Lu = auxx + 2buxy + cUyy + 2dux + 2ezs, = 0,
(2.7)
n
where a,b,c,d,e are continuous functions of X,Y in
with ac-b 2 >0,a>0.
[Hint: Prove first the maximum princip'le for solutions of Lu > 0, using that at a
maximum point in Q
lIyy .;;
o.
Apply this to u + ev where
v=exp[ M(x-xoi+(Y-Yoi)],
with (xo,Yo) outside Q and M sufficiently large.]
4. Let n = 2 and Q be the half plane X2 > O. Prove the maximum principle (2.3) for
u E C 2(Q) n Co(n) which are harmonic in n, under the additional assumption
that u is bounded in
(The additional assumption is needed to exclude
examples like u = X2') [Hint: Take for E > 0 the harmonic function
n.
U(Xb X2)- elogylX?+(X2+ 1)2 .
83
4 The Laplace equation
Apply the maximum principle to a region
Let e~O.]
Xr+(X2
+ 1)2 < a 2,X2;;' 0 with large a.
3. The Dirichlet Problem, Green's Function,
and Poisson's Formula
We derived the representation (1.23) for u in terms of Cauchy data,
involving a fundamental solution G(x,~. If here G(x,~=O for xEan and
~ E n, the term with du / dnx drops out and we have solved the Dirichlet
problem. We call a fundamental solution G(x,~) with pole ~ a Green's
function (for the Dirichlet problem for the Laplace equation in the domain
n), if
(3.1)
for XEQ, ~En, x~~, with K defined by (1.16), (1.15a,b), where v(x,~ for
~En is a solution of Axv=O, of class C 2(Q), for which
G(x,~)=O
for xEan, ~En.
(3.2)
To construct G in general we have to find a harmonic v with v = - K on
an, which is again a Dirichlet problem. However in some cases G can be
produced explicitly. This in particular is the case when n is a halfspace or
a ball; then G can be obtained by reflection, leading to Poisson's integral
formula. We note that for n=2 the Laplace equation is invariant under
conformal mappings. Thus if we can solve the Dirichlet problem for a
circular disk, we can solve it for any region which can be mapped
conformally onto a disk.
To derive Poisson's integral formula for the solution of the Dirichlet
problem for the ball of radius a and center 0,
n= B(O,a) = {xllxl < a},
(3.3)
we make use of the fact that the sphere an is the locus of points x for
which the ratio of distances r=lx-~1 and r*=lx-~*1 from certain points
~ and ~* is constant. Here for ~ we can choose any point of n. Then ~* is
the point obtained from ~ by "reflection" with respect to the sphere an,
that is,
(3.4)
One easily verifies that
r*
a
(35)
-;:-=m=const. for xEan.
For n > 2 the fundamental solutions with poles
I
2-n
K( x,o;;~) (2-n)w
n r,
84
~
K( ~*)_
X,o;;
-
and
~*
I
r*2-n
(2-n)wn
3 The Dirichlet Problem, Green's Function, and Poisson's Formula
are then related for x E a{l by
K(x,~*)= (
IDa )2-nK(x,~).
Thus the function
(3.6)
vanishes for x E a{l. Moreover the second term is singular only when
n, the function K(x,~*) is harmonic in x
throughout n. Thus G is a Green's function and formula (1.23) applies for
u E C 2(n). In the special case where ~u = 0 in the ball {l, and u E C 2(n) we
find, after a simple computation, Poisson's integral formula
x=~*. Since ~* lies outside
u(~)=
valid for
I~I
IIxl=aH(x,~)u(x)dSx
(3.7)
< a. Here H, given by
_ I a2_1~12
H(x,~) - - I I:.ln ,
aWn x-,.
(3.8)
is the Poisson kernel. We arrive at the same formula for n=2.
For given boundary values u = f on a{lJormula (3.7) solves the Dirichlet
problem, provided that problem has a solution U E C 2(n). We shall verify
directly that for f continuous on a{l the problem actually has a solution
given by Poisson's formula:
Let f be continuous for
for I~I = a and by
Ixl = a.
u(~)=
Then the function u(~) given by fm
IIxl=a H(x,~)f(x)dSx
(3.9)
for I~I < a, is continuous for I~I..;; a, and in Coo and harmonic for
1~I<a.
The proof follows easily from the following properties of H:
for Ixl";;a, 1~I<a,
(b) ~~H(x,~=O for 1~I<a, Ixl=a.
(c) flxl_aH(x,~dSx=1 for 1~I<a.
(d) H(x,~>O for Ixl=a, I~I <a.
(e) If Irl=a, then
(a)
H(x,~ECOO
lim
~-+~
x~~.
H(x,~)=O
IEI<a
uniformly in x for
Ix - r I> 8 >O.
85
4 The Laplace equation
Here properties (a), (d), (e) are clear by inspection of (3.8). Property (b)
follows either from (3.8) or by ·observing that G(x,~) as defined by (3.6) is
harmonic in x, that
(3.10)
G(x,~)= G(~,x),
and that H(x,~)=dG(x,~)/dnx. To derive (c) directly would be tedious;
but (c) follows simply by applying (3.7) to the function u(x) = 1, which is
harmonic and has boundary values 1.
By differentiation under the integral sign we find immediately from (a),
(b) that the u(~ defined by (3.9) belongs to C<X> and is harmonic for I~I < a.
There remains to prove the continuity of u for I~I..;; a. Let Ir I= a, I~I < a.
By (c)
uW-fG)= ~xl=aH(X,~)(j(x)-f(n)dSx
(3.11)
=/\ +/2 ,
where
f
f
... ,
Ix-~1<6
Ix-~1>6
Ixl=a
Ixl=a
For a given 10 > 0 we can choose 8 = 8 (e) > 0 so small that
If(x)-f(nl<e for Ix-rl<8, Ixl=a,
since f is continuous. Then lId";; 10 by (c), (d). Let maxi f(x) I= M for
Ixl = a. By (e) we can find a 8' such that
10
H(x,~)<
\ forl~-rl<8',lx-rl>8,
(3.12)
2Mwn a n where 8' depends on 10 and 8=8(10), and hence only on e. (See Figure 4.2.)
Then also 1/21 < e. Hence
lu(~)- f(nl <210
for I~- rl <8', I~I <a,
which shows that u is continuous at the boundary point r. This completes
the proof of the theorem.
As an application we derive estimates for the derivatives of a harmonic
function u in terms of estimates for u. We take at first the situation, where
~ is the balllxl < a, where u E C 2(n), and Au = 0 in ~. Using (3.7), (3.8) we
find by a simple computation that
(3.l3a)
and hence that
lu;(O)I..;;~ max lu(x)l.
,
86
a Ixl=a
(3.l3b)
3 The Dirichlet Problem, Green's Function, and Poisson's Formula
Figure 4.2
Let next au=O in any open set Q. Let gEQ and let g have distance d(~
from the boundary aQ. We apply (3.13b) to a ball of any radius a < d(~
with center at g. For a~dm we obtain the inequality
(3.13c)
Similar inequalities obviously hold for higher derivatives also. Let w denote
a compact subset of Q. Then there exists a positive lower bound for d(~ in
w. All harmonic U with a common upper bound M for lui in Q, will have a
common upper bound in w for the absolute values of their derivatives up
to a fixed order.
The inequalities for derivatives of u in terms of u lead to completeness
and compactness properties of the set of harmonic functions. Let there be
given a sequence Uk E C 2(Q) n Co(U) satisfying aUk = 0 in Q. Assume first
that the Uk converge uniformly on aQ towards a function f. Then the Uk
converge uniformly in n to a continuous function u by the maximum
principle. In any compact subset w of Q for a fixed multi-index a the
derivatives D "Uk converge uniformly to D "u. Hence u E C OO(Q) and au = O.
(We conclude in particular that uniform limits of harmonic functions are
again harmonic.) Under the weaker assumption that lukl..;; M on aQ for all
k, we have a common bound for the Uk and their first derivatives in the
compact subset w. By Arzela's theorem there exists then a subsequence of
the Uk which converges uniformly in w towards a function u. We conclude
as before that u is harmonic in w.
87
PROBLEMS
1. Verify the symmetry property (3.10) for the Green's function (3.6).
2. Let n=2 and 0 be the halfplane x2>0. For
~=(~l>~~
define by reflection
~.=(~l> -~~. .
(a) Show that G(x,O=K(x,O-K(x,~·) is a Green's function for O. Derive
formally the corresponding Poisson formula
u(~) = u(~ ,~ ) = 1.
I 2
'Ir
f
00
-00
~d(xl)
(xl-~1)2+~l
(3.14)
dx .
I
(b) Show that (3.14) actually represents a bounded solution of the Dirichlet
problem for 0, if j(XI) is bounded and continuous.
(c) Show that the maximum principle is satisfied by the solution (3.14) (see
problem 4 of Section 2).
3. For n=2 find a Green's function for the quadrant
reflection.
XI
>0, X2>0 by repeated
4. (Liouville's theorem.) Prove that a harmonic function defined and bQunded in
all of Rn is a constant. [Hint: Apply (3.13c) to balls of increasing radius.]
5. Let 0 be the halfspace Xn > O. If u E C 2(0) n COCO), ~u =0 in 0, u =0 on a~, and
u is bounded in 0, then u=O (In the "counter-example" U=Xn the function u is
not bounded). [Hint: Continue u as an odd function of Xn into all of lin.]
6. Letf((1) be a C 4-function of period 2'1r with Fourier series
00
f«(J)= ~ (ancos(n(J)+bnsin(n(J».
n=O
(a) Prove that
00
•
(3.15)
u= ~ (ancos(n9)+bnsin(n(J»rn
n=O
represents in polar coordinates r,(J the solution of the Laplace equation
= 0 in the disk X2 +y2 < 1 with boundary values f.
(b) Derive Poisson's integral formula (3.7), (3.8) for n=2 from (3.15) by substituting for the 0", bn their Fourier expressions in terms of f and interchanging summation and integration.
~u
7. (Harnack's inequality.) Let uEC 2 for
for Ixl < a. Show that for I~I < a
Ixl<a,
uEC o for
Ixl";a,
u~O, ~u=O
a n- 2(a -I~D u(O)..; u(~)..; a n- 2 (a+ I~I) u(O).
(a+I~Dn-1
(a-I~lt-1
(3.16)
8. Show that the constant n in the inequality (3.13b) can be replaced by
'In
2nwn _1
(n-l)wn
(3.17)
and that this is the smallest possible constant in the inequality. [Hint: Use
u=lxnl on Ixl=a in (3.13a).]
88
4 Proof of Existence of Solutions for the Dirichlet Problem
9. Let Au(x) = 0 and lu(x)l.;;;; M for
(a) Show that
Ix-~I <a.
(3.18)
where Yn is defined by (3.17) for dimension n. [Hint: Apply (3.13c) successively to the kth derivatives in the balls Ix-~I.;;;; a(m-k)/m for k=
O,l, ... ,m-l.]
(b) Show that u(x) is repres..nted by its series in terms of powers of x-g for
(3.19)
4. Proof of Existence of Solutions for
the Dirichlet Problem Using Subharmonic Functions
("Perron's Method")
The proof is based on the fact that we can characterize the solution w of
the Dirichlet problem by an extremal property; we only have to prove that
a function with the extremal property exists. Assume we had a function
WEC 2(U)n CO(Q), such that aw=O in U and that w takes the prescribed
values j on au. Let u be any function which is subharmonic in U and
continuous in Q, for which u <J on au. Then u - w .;;;; 0 on au, and hence
u - w .;;;; 0 in U by the maximum principle for subharmonic functions on p.
82. Thus the value of the solution w of the Dirichlet problem at any point
~ of U is the largest value taken at ~ by any subharmonic function with
boundary values .;;;; j, and this will be our extremal property. The construction of w from its extremal property proceeds by a number of easily
verified elementary lemmas. Essential use is made of the maximum principle and of the solvability of the Dirichlet problem for a ball, assured by
Poisson's integral formula (3.7), (3.8). This somewhat limits the extension
of the method to more general partial differential equations.
In what follows we denote again by B(~,p) the open ball of center ~ and
radius p in Rn , by jj(~,p) its closure, and by S(~,p) its boundary. For a
continuous u = u(x) we denote by
-I.
i-n
Mu(~'p)=-P
Wn
S(~,p)
u(x)dSx
(4.1)
the "arithmetic mean" of u on S(~,p). We consider an open, bounded, and
connected set U in Rn. In accordance with our previous definition we call u
subharmonic in U, if u E CO(U), and if for every ~ in U the inequality
(4.2)
holds for all sufficiently small p. We denote by o(U) the set of functions
subharmonic in U. The maximum principle proved on p. 82 constitutes
89
4 The Laplace equation
Lemma I. For uEO'(O)n CO(n) we have
(4.3)
maxu";; maxu.
n
an
Definition. For uECo(O) and jj(~,p)cO we define Ut;,p as that function in
CoCO) for which
u€,p(x)=u(x)
axu€,p(x)=O
for xEO, xtlB(~,p)
(4.4a)
(4.4b)
for xEB(~,p).
(That is, u~,p is obtained from U by replacing U in the ball B(~, p) by the
harmonic function that agrees with U on the boundary of the ball. The
existence and uniqueness of u has been established earlier.)
Lemma II. For uEO'(O) and jj(~,p)EO we have
u(x)..;; u€,p(x)
(4.5a)
for all x EO
(4.5b)
u€,p EI.J(O).
PROOF. By definition (4.5a) holds for x tl B(~, p), with the equal sign. With
u also its restriction to B(~,p) is subharmonic in B, and so is the harmonic
funct~n - uE,p' Thus u- uE,p EO'(B(~,p». Since u- uE,p is continuous in the
ball B(~,p) and vanishes on its boundary, it follows from Lemma I that
u- uE,p";; 0 in B(~,p), which establishes (4.5a). In order to prove (4.5b) we
have to show that for any ~EO
uE,p(O..;; Mu!,p (r,'T)
r
(4.6)
for all sufficiently small 'T. Now this is true for tljj(~,p) by (4.4a), since
uEO'(O), and is true for EB(~,p) since then uE,p is harmonic near There
remains the case where ES(~,p). But then by (4.2), (4.5a)
r
r
Ug,p(r) = u(r)..;; Mu(r,'T)";; Mu!,p (r,'T)
r.
(4.7)
for all sufficiently small 'T.
Lemma
m. For UE 0'(0) inequality (4.2) holds whenever ii(~, p) cO.
PROOF.
By (4.4a), (4.5a) and the mean value theorem for harmonic func-
tions
Lemma IV. A necessary and sufficient condition for U to be harmonic in 0 is
that both u and - u belong to 0'(0).
PRoOF. The necessity is implied by the mean value theorem for harmonic
functions. Let, on the other hand, u E 0'(0), - u E 0'(0). Then for jj(~, p) cO
90
4 Proof of Existence of Solutions for the Dirichlet Problem
by Lemma III
U(x)..;; U~,p(x),
-U(x)";;-U~,p(x)
forxEn.
D
Thus u=u~,p; hence U is harmonic in B(~,p).
Lemma IVa. If UE Co(n), and if for each ~En
u(~)=Mu(~'p)
for all sufficiently small p, then U is harmonic in
PROOF.
n.
The assumptions imply that UE o(n), - UE o(n).
D
Definition. For f E CO(an), define
OJ(Q) = { ulu E CD(Q) n o(n), u";;f on an}
(4.8)
wix)= sup u(x) for xEn.
(4.9)
UEOf
(£!)
Set
m=inff,
(4.10)
f.L=supf·
Wf?.. observe that the constant m belongs oIQ), and hence that the set
oln) is not empty. Also by Lemma I
(4.11)
u(x)..;; f.L for UEOj(Q), xEQ.
Thus Wj is well defined.
Lemma V. Let u1"",ukEoIQ) and v=max(u1,,,,,Uk)' Then vEo/Q).
PROOF. One easily verifies that vECD(Q), since u1"",UkECD(Q). Then by
(4.2) for any ~En and all sufficiently small p.
v(~) =max(ul(g), ... , Uk(~»
..;; max(MUl(~'P), ... ,MUk (~,p»)..;; Mv(~'p)·
Lemma VI. Wj is harmonic in
D
n.
Let jj(~,p)cn. Let xI,x 2 , ... be a sequence of points in B(~,p')
where p'<p. By (4.9) we can find functions uk EOj(Q) for k,j=I,2,3, ...
such that
PROOF.
(4.12)
Relation (4.12) is preserved if any
>ufc in n, since then
ui is replaced by any
U
E o/Q) for which
U
91
4 The Laplace equation
Define the sequence uj(x) for xE~,j= 1,2, ... by
uj(x) =max( u{(x), ... ,uj(x)).
Then ~ E a/~) by Lemma V and uj(x) ~ u[(x) for x Efl,j ~ k. Thus
~imuj(Xk)=wf(Xk)
forallk.
(4.13)
J-+OO
Replacing, if necessary, ~ by max(uj,m), we can bring about that
m~~(x)~ IL
u
forxEfl
ul.
(4.14)
(4:.! 1». Finally replacing j by
p , we can arrive at a sequence
ul E alfl), for which (4.13), (4.14) hold, and which are harmonic in B(E,p).
Since the uj lie between fixed bounds, we conclude from the compactness
property of harmonic functions (see p. 87) that there exists a subsequence
of the uj(x) converging to a harmonic function W(x) for x in the compact
subset B(E,p') of B(E,p). It follows from (4.13) that
(s~e
wf(xk)=W(x k )
for all k.
(Observe that here the harmonic function W could depend on the choice
of the sequence Xk and on that of the subsequence of the js.) Taking first
for the Xk a sequence converging to a point x in B(E, p'), we conclude from
the continuity of the corresponding W that limk~oowlxk) always exists,
hence that wf is continuous in B(E,p'). Taking next for the Xk a sequence
dense in B(E, p') we find that Wj agrees with a harmonic function W in all
x k, and hence, by continuity, in all x E B(E, p'). This means that wf is
harmonic in a neighborhood of E, and hence throughout fl.
D
The harmonic function wf is our candidate for the solution of the
Dirichlet problem. We have to show that it has the prescribed boundary
values j on afl. This requires some additional assumption on the nature of
the boundary afl of fl. The assumption is formulated conveniently as
existence of certain barrier junctions, that is of subharmonic functions that
are zero at one boundary point and negative at all others.
Barrier Postulate. Let there exist for each 7J E afl a function ("barrier
function") Q'l(x) E Co(~) n a(fl), for which
Q'l(7J) =0,
Q'l(x)<O forxEafl,x*7J.
(4.15)
Lemma YD. For 7J E afl
(4.16)
PROOF.
Let e and K be positive constants. Then
u(x)= j(7J) - e+ KQ'l(x)
92
(4.17)
4 Proof of Existence of Solutions for the Dirichlet Problem
belongs to CO(O)n o(~), and satisfies
u(x)<f(.,,)-e forxEa~,
u(.,,) = f(.,,)-e.
Since f is continuous there exists a 8 = 8(e) > 0 such that f( x) >f(.,,) xEa~, Ix-.,,1<8. Then by (4.15), (4.17)
f
for
u(x) < f(x)
(4.18)
for x Ea~, Ix-."I < 8. Since Q.,,(x) has a negative upper bound for Ix-."I;;>
8, we can find a K = K( e) so large that (4.18) holds as well for x E a~,
Ix -.,,1;;> 8. Thus u E (10) and consequently
u(x) < Wj(x)
for xE~.Then also
f(.,,)-e= lim u(x) <liminfw/x).
x-'>."
xEs:!
o
For e~O we obtain (4.16).
Lemma vm. For." E a~
(4.19)
PROOF.
In view of Lemma VII it is sufficient to show that
lim sup w/x) < f( .,,).
(4.20)
xEs:!
We consider the function -
W
_/x) which is defined in
~
by
-w_/x)= --:supu(x) foruEa_j(O).
Writing u = - U we have
- W_j(x)=inf U(x)
(4.21)
taken over all U for which
-U<
-f
ona~.
(4.22)
Then for any u E 17/0) and U satisfying (4.22)
u-U<O
on a~, and hence in ~ by Lemma I. Thus also by (4.9), (4.21)
W/x) < -w-ix) forxE~
Applying Lemma VII to
W
_Ix) we have then
limsupw/x) <lim sup ( - W_j(x») = -liminfw_j(x) < f(.,,)·
xEs:!
x-'>."
xEs:!
x-'>."
xEs:!
0
x-'>."
Lemma VIII implies that wjE CO(O) when we define Wj= f on a~. We thus
have proved:
93
4 The Laplace equation
Theorem. If the domain !J has the postulated barrier property there exists a
solution wE C 2(!J) n CO(Q) of the Dirichlet problem for arbitrary continuous boundary values f.
The barrier postulate can be verified for a large class of domains !J.
Take, for example, the case where the open set is strictly convex in the
sense that through each p~int 1] of a!J there passes a hyperplane '1TTJ having
only 1] in common with !J, We then can use for QTJ(x) a suitable linear
function which vanishes on '1TTJ and is negative on Q except at the point 1].
See also the following problem.
PROBLEM
Show that the open set n satisfies the barrier postulate if for each 1/ E an there
exists a ball B(~, p) such that 1f(~, p) and Q have just the point 1/ in common. [Hint:
Use for - QTJ(x) a fundamental solution with spherical symmetry about the point
~.]
5. Solution of the Dirichlet Problem
by Hilbert-Space Methods*
A variety of methods have been invented to solve the Dirichlet problem for
the Laplace equation. Of greatest significance are methods that make little
use of special features of the Laplace equation, and can be extended to
other problems and other equations. Among these is the method to be
described now, reducing the Dirichlet problem to a standard problem in
Hilbert space, that of finding the normal to a hyperplane. Typically the
solution of the Dirichlet problem proceeds in two steps. In the first step a
modified ("generalized") Dirichlet problem is solved in a deceptively simple
manner. The second step consists in showing that under suitable regularity
assumptions on region and data the solution of the modified problem
actually is a solution of the original problem. The second step, which
involves more technical difficulties, will not be carried out here in full
generality. From the point of view of applications one might even take the
attitude that the modified problem already adequately describes the physical situation.
In this book we can only summarize the relevant properties of a Hilbert
space. We start with the broader concept of a vector space S (over the real
number field). This is a set of objects u, v, . " closed under the operations
of addition and of multiplication with real numbers ("scalars") A, p" ••••
For u,vES and scalars A,p, we have AU+p,V defined as an element of S
and obeying the usual arithmetic laws. The simplest example is the set of
vectors in the finite-dimensional space IRn.
We next assume that in S there is defined for u, v E S a real-valued
* ([9], [14], [10], [llD
94
5 Solution of the Dirichlet Problem by Hilbert-Space Methods
inner product (u, v), such that
(5.1)
(u,v) = (v,u),
(Xu + /LV, w)=X(u, w)+ /L(v, w)
(0,0)=0,
(5.2)
(u,u»o for u'f'O,
where "0" denotes both the O-element of S and the real number. (This
assumption characterizes S as an "inner product space".) Property (5.2)
suggests defining the length ("norm") of a vector u by
lIull=Y(u,u) .
One easily verifies then the Cauchy and triangle inequalities
l(u,v)lo;;;;lIullllvll,
A sequence
limit uE S if
u\u 2 , •••
lIu+vllo;;;;lIull+llvll.
(5.3)
(5.4)
ES is said to converge to the (necessarily unique)
lim lIu-ukll=O.
k-4OO
A sequence u \ u2 , ••• E S is called a Cauchy sequence if
lim lIu k - uill =0.
j,k-400
(5.5)
(5.6)
The space S is complete and is called a Hilbert space if every Cauchy
sequence in S converges, (to an element of S). Every inner product space S
can be completed, that is imbedded into a Hilbert space H in which S is
dense. This completion is achieved in analogy to the construction of real
numbers from rational ones. We define H as the set of Cauchy sequences
{U i ,U 2 , ••• } in S, identifying Cauchy sequences {u\u 2 , ••• } and {V i ,V 2 , ••• }
for which
lim lIuk-vkll=O.
k-4OO
(5.7)
We define multiplication by a scalar, addition, and inner product of
elements of H by performing these operations for each element of the
corresponding Cauchy sequence. It is easily seen (as for real numbers) that
the resulting H is indeed complete ap.d that S is dense in H; (as a matter of
fact an element of H given by the Cauchy sequence u\u 2 , ••• ES is just the
limit of the Uk).
A linear functional cp on an inner product space S is a real-valued
function defined on S for which
cp(Xu + /Lv) =XCP(u) + /LCP( v).
(5.8)
The functional is bounded if there exists an estimate
Icp(u)lo;;;; Mllull
(5.9)
valid with the same M for all u E S. For any V E S the inner product
(u,v)=cp(u) defines a bounded linear functional by (5.1), (5.4). In a Hilbert
space H the converse holds as well:
95
4 The Laplace equation
Representation theorem. Every bounded linear junctional cp on H can be
represented uniquely in the form cp(u)=(u,v) with a suitable element v of
H.
Associating with a functional cp, (that does not vanish identically), the
hyperplane 'fT in H consisting of the points u with cf>(u)= I, we can think of v
as giving the direction of the normal to 'fT, since for any two points uI,u 2 in
'fT the scalar product (u I - u 2 , v) vanishes. Interpreting the direction of v as
that of the normal to the plane 'fT, suggests that v also has the direction of
the shortest line from the origin to 'fT. Indeed the point
v
v
w=--=-(5.10)
cp( v )
lies on
'fT,
and is the point of
'fT
( v, v)
closest to 0, since for any u on 'fT
lIu11 2= Ilu- w+wI1 2 = IIw112+ lIu-wI1 2 +2(u-w, w) ~ IIwl12 (5.11)
since
(u- w,w)= (u- w,v)
(v,v)
cp(u)-cp(w)
(v,v)
O.
This extremal property of w or v is made use of in the standard proof of
the representation theorem. (See Problem 1.)
We now reformulate the Dirichlet problem for the Laplace equation as
the problem of representing a certain bounded functional cp in a Hilbert
space as an inner product (u, v). In the usual version of the problem one
looks for a function U with domain n satisfying Il U = 0 in 0 and U =f on
ao. Assuming the prescribed f to be defined not only on ao but throughout
n, we look instead for the function v = U - f for which
on ao,
Ilv= -w in 0
(5:12)
with w=llf given. For simplicity we restrict ourselves to an open bounded
connected set 0, to which the divergence theorem applies. In the space of
functions of class Cl(n) we define the bilinear form (u,v) by
v=O
(u,v)=
Jrg ~k Ux•Vx• dx.
(5.13)
With this definition the space C I(n) is not an inner product space, since
(u,u)=O has the non-vanishing solutions u=const. Denote by ed(n) the
subs.(>ace of functions u in ~ I(n) that vanish on
Obviously (u, v) can be
used· as inner product on
by the Dirichlet integral
an.
cd with the corresponding squared norm given
(5.14)
Let v E C 2(n) be a solution of (5.12), where the prescribed w belongs to
96
5 Solution of the Dirichlet Problem by Hilbert-Space Methods
CO{U). Then for any u E CJ{U) we have by the divergence theorem
(u,v)= -
L
ullvdx= Luwdx.
(5.15)
This suggests that v can be found by simply representing the known linear
functional
cf>(u) =
f uwdx
(5.16)
as an inner product (u,v). To make use of the representation theorem we
have to complete CJ{O) into a Hilbert space HJ{O) with respect to the
Dirichlet norm (5.14), and to prove that the cf>{u) defined by (5.16) gives rise
to a bounded linear functional in that space. Our modified version of the
Dirichlet problem is then the following:
Find a v E H J{O) such that
(u,v)=cf>(u) for all uEHJ(O),
(5.17)
where (u,v) and cf> are defined by (5.13), (5.16).
To show that the functional cf> is bounded, we have to derive an
inequality (5.9). Since by Cauchy
(~UWdx
r<L
u 2 dx
~ w dx,
2
(5.18)
it is sufficient to show that there exists an N such that
(5.19)
We show this Poincare inequality first for u E CJ{U). Since 0 is bounded it
can be enclosed in a cube
f:
Ix;l<a
fori=I,2, ... ,n.
We continue u as identically zero outside O. Then for any x={x,,. .. ,xn)E
f
Thus
97
4 The Laplace equation
Integrating over
X 2 , • •• ,
xn from - a to a we find
r
r
(S.19a)
2
2
2
Jr u dx <:.4a Jr uXl dx '
which implies (S.19) with N =4a 2 • An element u of HJ is represented by a
Cauchy sequence U 1,U 2, ••• E CJ(Q) for which
Ilu k - ujll =0.
lim
j.k~oo
By (S.19), this implies that also
r(uk-ujfdx=O.
lim
(S.20)
j.k~oo Jo
By (S.18), (S.16) the numbers </>(u k ) then form a Cauchy sequence, and we
can define </> for the element u of HJ by
</>( u) = lim </>( Uk).
(S.21)
k~oo
Since also
lIull = lim Ilukll
k~oo
by definition, the inequality (S.9) for u E CJ(Q) implies the same inequality
for u E H J. One finds then that </> can be extended to the Hilbert space H J
as a bounded linear functional. The representation theorem guarantees the
existence of a v in HJ(U) for which (S.17) holds, and thus solves our
modified Dirichlet problem. *
As mentioned earlier there remains the task of identifying the solution
vEHJ(U) of the modified problem with a function v that satisfies (S.12) in
the ordinary sense. Here the verification of the differential equation
Llv = - w, say for wEe 1(Q), is not as difficult as to show that v =0 on au.
One first observes that the solution v of the modified problem is representable by a Cauchy sequence vl,v 2 , ••• in CJ(Q), with respect to the norm
(S.14). This implies by (S.20) applied to the v k that v can be identified with
a function which is square integrable in the sense of Lebesque in U, and for
which
(S.2Ia)
Using for u a test function of class COO(U) and of compact support we have
(v,u)= .lim (vj,u)=?-m
J~OO
J~OO
rL v~
Jo
k
k
Ux
dx
k
= - .lim IvjLludx= -lvLludx.
J~OO
0
0
* The modified problem can be solved more generally for any bounded open set 0 by defining
HJ(O) as the completion with respect to the Dirichlet norm of the set of all u E C 2(O) that
have compact support in O. This does not involve any regularity assumptions on a~.
98
5 Solution of the Dirichlet Problem by Hilbert-Space Methods
Since test functions belong to
Hd it follows from (S.16), (S.I7) that
! vll.udx= - ! wudx.
(S.22)
Hence v is a solution of the P.D.E.
(S.23)
ll.v= - w
in the sense of distributions (see p. 69). From this we can prove that
v E C2(~) and ll.v = - w. Take any z E~ and p so small that the ball
B(z,3p) lies in ~. Take any test function <p with support in B(z,p) and a
fixed test function r(x) which has the value I in B(z,2p). Set
(S.24)
where K is the fundamental solution given by (1.16). By Poisson's formula
(1.28)
where
Thus
J v(x)<p(x)dx=! v(x)<pI(x)dx+ J v(x)<p2(x)dx.
(S.2Sa)
Here, since r(x)u(x) again is a test function, by (S.22)
J v(x)<pI(x)dx= - J w(x)nx)u(x)dx
(S.2Sb)
<P2(X)=ll.xJ (1- t(x»K(x, g)<p(g) dg
=
f.
B(z,p)
F(x,g)<p(g)dg.
(S.2Sc)
Here
F(x,g)=ll.xC(I - nx»K(x,g»
belongs to Coo in x,g for gEB(z,p) and all x, since K(x,g) is singular only
for x =g while 1- r(x)=O in B(z,2p). Moreover
Il.~F(x,g)=O,
(S.26)
since Il.EK(X,g) =0 for x+g. We find from (S.24), (S.2Sa,b,c) that
! v(g)<p(g)dg= J<p(g)dg(J (- w(x)t(x)K(x,g) + F(x,g)v(x»dx.)
Since this identity holds for all test functions <p with support in B(z,p), we
conclude that for almost all gEB(z,p)
v (g) = - J K(x,g)nx)w(x)dx+ J F(x, g)v(x) dx.
99
4 The Laplace equation
It follows from (5.26) and Poisson's equation (1.28) that v coincides almost
everywhere* in B(z,p) with a function of class C 2 for which
~Ev(~) = - r(~)w(~) = - w(~).
Since z is arbitrary we have ~v = - w throughout n.
The proof that v has boundary values 0 will be given here only for the
dimension n=2. Prescribe a number e>O. To v there corresponds a
Cauchy sequence v k E
We can find a number j such that
eden).
Ilvj-vkll,e
e
for all k>j. Denote by d(~ for ~En the distance of ~ to a closest point
of the boundary curve an. We assume that an is sufficiently regular and
d(~)=I~*-~1 sufficiently small. Then each x in B(~,d(~) can be joined to a
point of an by a segment parallel to ~~* and of length '4dm. The union
U of these segments covers B(~,d(~» (See Figure 4.3). Poincare's inequality
(5.19a) applies* to u=Vj-V k in U and yields that
f f (v j - Vk)2dxl dX2' 16d2(~)lIvi _v k Il 2 , 16d2(~)e2
u
for k >j. By (5.21a) then
f f
(vi - V)2 dx 1 dx 2, 16d2(~)e2.
(5.27)
B(E,d(~)
For the solution v of
~v =
- w we have by (1.24)
v(~)= - f f (1f(lx-~I)-1f(p»W(X)dXldx2+-21
~~
r
~~~~
v{x)ds,
(5.28)
where.1f(p)=(1/2'1T)logp. Let M=maXnlwl. Multiplying (5.28) by p and
integrating with respect to p from 0 to d(~ we find that
'1Td2{~)lv{~)I' f f
Ivldx+O(M d 4m)
B(E,d(~)
,
f f
Iv-vjldx+
B(E,d(~»
ff
Ivjldx+O(Md4{~».
B(~,d(m
Since the continuous function vi vanishes on an we have Ivi(x)1 < e for
x E B(~,d(~) provided dm is sufficiently small. In addition by (5.27)
( flv-dl dX)2 ''1Td 2{O
B(E,dm)
f f
(v-vjidx'16'1Td4{~)e2.
B(~,d(~»
• As a distribution the values of v are only determined "almost everywhere."
tIn the proof of (S.l9a) we did not need that u vanishes everywhere on the boundary of n. It
is sufficient to know that in n each coordinate has a range of length .;;; 2a, and that each point
of n can be joined in n by a parallel to the XI-axis to a point where u vanishes.
100
5 Solution of the Dirichlet Problem by Hilbert-Space Methods
Figure 4.3
It follows then that for sufficiently small
d(~
v(~) =O(e+ M d2(~)).
This implies that
v(~)
tends to 0 as
~
approaches the boundary
a~.
PROBLEMS
1. Prove the representation theorem in Hilbert space H by finding a point in a
plane with the minimum distance from the origin. [Hint: Let m=infllull for
q,(u)= 1. Take a minimizing sequence w k with </>(w k) = 1, IIwkll~m. From
k
m2~11 w
;w' r=-}lIwkIl2+-}1I~1I2-illwk-~112
.
prove that the w k form a Cauchy sequence converging to an element w of H
with q,(w)= 1, IIwll = m. Show that (w,u)=O for q,(u)=O, and finally that
(v,u)=q,(u) for v=w/(w,w) and all u.J
101
4 The Laplace equation
2. Show that a solution U E C 2(n) of
~U=O inn,
U=J onan
minimizes the Dirichlet integral (U, U) among all functions in C 1(fi) with
boundary values J ("Dirichlet's principle"). Show that the corresponding modified Dirichlet problem consists in finding a v satisfying (5.17) for the functional
q,(u) =(u,f).
3. Consider the Dirichlet problem for n = 2 with n being the unit disk referred to
polar coordinates r,O. Let J(O) denote the boundary values assumed to be
continuous, and u(rcosO, rsinO) the solution of ~u=O with boundary valuesJ,
known to be of class C 2(n) n Co(n).
(a) Show that for r< 1
00
u= ~ (akcoskO+bksinkO)rk,
k=O
(5.29)
where ak,bk are the Fourier coefficients of J. [Hint: prove first that u has a
Fourier expansion of the form (5.29) for r< I with certain anobn • Show that
by continuity the anobn are the coefficients of J.]
(b) Show that the Dirichlet integral of u is given by
(5.30)
(c) Show that there are continuous J for which the Dirichlet integral for the
corresponding u is infinite. [Hint: Find sequences ak, bk for which the series
in (5.30) diverges, while the series
00
~ (Iakl + Ibkl)
k=\
converges.] (This shows that the Dirichlet problem cannot be solved for all
continuous boundary values by the Hilbert-space approach.)
4. (Best constant 1fA in Poincare's inequality). Show that if there exists a function
u E C 2(n) vanishing on an for which the quotient
(u,u)
fDf u dx
2
reaches its smallest value A, the~ u is an eigenfunction to the eigenvalue A, so
that ~U+AU=O in n. In fact A must be the smallest eigenvalue belonging to an
eigenfunction in C 2(n).
102
Hyperbolic equations in
higher dimensions*
5
1. The Wave Equation in n-dimensional Spacet
(a) The method of spherical means
The wave equation for a function u(x\, ... ,xn,t)= u(x,t) of n space variables X\""'Xn and the time t is given by
Ou=Utt -C2AU=0
(1.1)
with a positive constant c. The operator "0" defined by (1.1) is known as
the D' Alembertian. For n = 3 the equation can represent waves in acoustics
or optics, for n = 2 waves on the surface of water, for n = 1 sound waves in
pipes or vibrations of strings. In the initial-value problem we ask for a
solution of (1.1) defined in the (n + I)-dimensional half space t > 0 for
which
u= f(x), ut=g(x)
for t=O.
(1.2)
The initial-value problem (1.1), (1.2) can be solved by the method of
spherical means due to Poisson. We associate generally with a continuous
function h(x)=h(x\, ... ,xn) in IR n its average Mh(x,r) on a sphere with
center x and radius r:
(1.3)
Setting y
= x + r~ with
I~I
= 1, we get
Mh(x,r)=-.!.. (
Wn
J1g1 = I
h(x+r~)dS~.
(1.4)
*([15]. [19D
t([2]. [6]. [17D
103
5 Hyperbolic equations in higher dimensions
Originally Mh(x,r) is defined by (1.3) only for r>O. We can extend its
definition to all real r using (1.4). The resulting Mh(x,r) then is an even
function of r, since replacing r by - r in (1.4) can be compensated for by
replacing the variable of integration ~ by -~. It is also clear from (1.4) that
Mh E C S (Rn+ 1) for h E C S (Rn) since we can differentiate under the integral
sign. For h E c 2(Rn) we find from (1.4), using the divergence theorem (4.1)
of Chapter 3 that
a
I
-ar Mh(x,r)= -wn )Ilil
r = 1 i~= 1 hx;(x+r~)~idSIi
n
=~
Wn
r
)11;1<1
axh(x+r~)d~
MUltiplying by r n - I and differentiating with respect to r yields
;r(r n- I ;rMh(x,r»)=axrn-IMh(x,r).
(1.5)
Thus the spherical means Mh(x,r) of any function hE c 2(Rn) satisfy the
partial differential equation
a) Mh (x,r) =axMh(x,r)
( ara + -n-I
- -a
r
r
2
-2
(1.6)
known as Darboux's equation. Using that the solution Mh(x,r) of (1.6) is
even in r, we find for its initial values
( aa Mh(x,r»)
Mh (x,O)=h(x),
r
r=O
=0.
(1.7)
Forming spherical means we can transform the initial-value problem for
the wave equation into one for a hyperbolic equation in two independent
variables. Let u(x,t) be a solution of (1.1), (1.2) of class C 2 in the half
space x ERn, t > O. We form the spherical means of u as a function of x:
Mu(x,r,t)=
~
Wn
r
)1~1=1
u(x+r~,t)dSIi.
(1.8)
Obviously u can be recovered from M u , since
Mu(x,O,t) = u(x,t).
104
(1.9)
1 The Wave Equation in n-dimensional Space
By (1.6)
!1xMu =
a) Mu'
( ara2 + -n-l
- -a
r
r
-2
On the other hand by (1.1), (1.8)
!1xMu=~
Wn
r
JI~I= I
!1xu(x+r~,t)dS~
1 a -1
=2"-2
2
c at
Wn
i
a2
1~1=1
1
u(x+r~, t)dS~=2"-2Mu'
c at
Hence Mu(x,r,t) as a function of the two scalar variables r,t for fixed x is
a solution of the P.D.E.*
(1.10)
The P.D.E. (1.10) depending on the parameter n (here equal to the
dimension of x-space) is known as the Euler-Poisson-Darboux equation.
Our Mu as function of r,t by (1.2), (1.8) is a solution of (1.10) with the
known initial values
(1.11 )
The initial-value problem (1.10), (1.11) can be solved most easily when
the number of space dimensions is n = 3t
Indeed by (1.1 0)
(1.12)
Thus rMu(x,r,t) as a function of r,t is a solution of the one-dimensional
wave equation with initial values
(1.13)
• The expression
02
n-l 0
-+--or2
r or
has been encountered already (Chapter 4, (1.9)) as the Laplacian of a function in R"
depending only on the distance r from a fixed point. Equation (1.10) thus asserts that the
spherical means of a solution of (1.1) on spheres with center x again form a solution of (1.1).
This is to be expected since the spherical means could plausibly be obtained by rotating the
solution u in all possible ways about x and averaging over all rotations. Since the wave
equation is invariant under rotations this procedure should lead again to a solution.
tThis is more difficult for other values of n. See problem 2.
105
5
Hyperbolic equations in higher dimensions
Thus by our general formula «4.13) of Chapter 2)
rMu(x,r,t) =
"2I [(r+ct)Mix,r+ct)+(r-ct)Mix,r-ct)]
Using that MtCx,r) and Mg(x,r) are even in r we are led to
Mu(x,r,t) =
(ct + r)Mix,ct+ r) - (ct - r)Mix,ct- r)
2r
I
+ -2
f
ct + r
~Mg(x,~)d~.
rc ct-r
Letting r tend to 0 and replacing differentiation with respect to r by
differentiation with respect to ct, we find by (1.9) that
u(x,t)= tMg(x,ct) + :t (tMix,ct»)
= ~I
g(y)dSy+ .l.(~I
4'1TC t Iy-xl=ct
at 4'1TC t Iy-xl=ct f(Y)dSy ). (1.14)
Any solution u of the initial-value problem (1.1), (1.2) of class C 2 for
t>O in n=3 space dimensions is given by formula (1.14), hence is unique.
Conversely for any f E C3(~3) and g E C2(~3) the u(x, t) defined by (1.14)
is of class C 2 and satisfies (1.1), (1.2). Indeed (1.2) follows by inspection,
using (1.7). Moreover by (1.6) for n=3 and r=ct
a2
a2
- (tMg (x,ct») = c -2 (rMg (x,r») = cr!l.xMg (x,r) = c 2!l.A tMg (x, ct».
at 2
ar
Thus tMg(x,ct), and similarly (a/at)tMj(x,cO, satisfy the wave equation
(1.1)* .
Formula (1.14) displays the relevant features of the solution u of the
initial-value problem for the wave equation in the case n = 3. First of all,
writing our spherical means in the form (1.4). we can carry out the
I-differentiations under the integral sign, arriving at the expression
U(X,t)=~f
4'1TC t
(tg(y)+f(y) +
Iy-xl=ct
~.iy.(y)(yj-X;))dSy.
i
•
(1.15)
(1.15) indicates that u can be less regular than the initial data. There is a
possible loss of one order of differentiability: u E e, Ut E C s + 1 initially,
guarantee only that u E c s - 1, ut E C S at a later time. This is the focussing
effect, present when n> 1t. For example the second derivatives of u could
.
.
·Incidentally the first expression for u in (1.14) defines u as a solution of (l.l) for all x,t,
since Mj(x,ct), Mg(x,ct) are defined for all t as even functions in t.
t Irregularities in the initial data are "focussed" from different localities into a smaller set,
("caustic") leading to stronger irregularities. This phenomenon does.not occur for n= I where
u is no worse than its data, as shown by formula (4.13) of Chapter 2.
106
1 The Wave Equation in n-dimensional Space
become infinite at some point for t > 0, though they are bounded for t = o.
In contrast to the pointwise behavior of u, we shall find that in the L2-sense
U does not deteriorate. This follows from the fact that the energy norm of U
E(t)=i J J
J(ut(X,t)+C2~U;i(X,t))dX
(1.16)
does not change at all with t. Indeed
c;: = J
( UtUtt + c2
~ Ux,Uxit ) dx = J
=0
( utDu + c 2
~ (UtUx,) Xi) dx
(1.16a)
if u(x,t)=O for all sufficiently large IxiAccording to (1.15) the value u(x,t) depends on the values of g and ofJ
and its first derivatives on the sphere S(x,ct) of center x and radius ct.
Thus the domain oj dependence for u(x,t) is the surface S(x,ct). (See
Figure 5.1.) Conversely the initial dataJ,g near a pointy in the plane t=O
only influence u at the time t in points (x, t) near the cone Ix - yl = ct. (See
Figure 5.2.) Let J,g have their support in a set U E 1R3. In order that
u(x, t)+O the point x has to lie on a sphere of radius ct with its center y in
U. The union of all spheres S(y,ct) for y EU contains the support of u at
the time t. This gives rise again to Huygens's construction for a disturbance confined originally to U. (See· p. 30.) The support of u spreads
With velocity c. It is contained in the region bounded by the envelope of
the spheres of radius ct with centers on au. Actually the support of u(x, t)
can be smaller. Take, for example, for the region U containing the support
ofJ,g the ball B(O,p) of radius p and center o. Then S(x,ct) for ct>p will
have a point in common with U only when x lies in the spherical shell
bounded by the spheres S(O,ct+p) and S(O,ct-p). For any fixed x and all
sufficiently large t (namely t> (lxl + p)/ c) we have u(x, t) =0. A disturbance originating in B (0, p-) is confined at the time t to a shell of
Figure 5.1
107
5 Hyperbolic equations in higher dimensions
~
~
I
t=O
Figure 5.2
u=O
t
=0
Figure 5.3
thickness 2p expanding with velocity c. (See Figure 5.3.) This accounts for
the .possibility of "sharp" signals being transmitted in accordance with
equation (1.1) in three dimensions. This phenomenon is due to the fact that
the domain of dependence for u(x,t) is a surface in x-space rather than a
solid region ("Huygens's principle in the strong form"). For most hyperbolic equations (even for the wave equation in an even number of dimensions) the principle does not hold. Disturbances propagate with finite
108
1 The Wave Equation in n-dimensional Space
speed but after having reached a point never die out completely in a finite
time at that point, like the surface waves arising from a stone dropped into
water.
While the support of the solution with initial data of compact support
expands, the solution decays in time.* Assume that j,g and the first
derivatives of j are bounded, and vanish outside B(O,p). Contributions to
the integral in (1.15) arise only from that portion of the sphere S(x,ct) that
lies inside the ball B(x,p). Elementary geometry shows that the area of
intersection of any sphere in 3-space with a ball of radius p is at most 4'1Tp2.
Thus the integral is at most equal to the maximum of the absolute value of
the integrand multiplied by 4'1Tp2. It follows that u for large t is at most of
the order of 1/ t.
PROBLEMS
1. (a) Show that for n=3 the general solution of (1.1) with spherical symmetry
about the origin has the form
F(r+ct)+ G(r-ct)
u=------r
r=lxl
(1.17)
with suitable F, G.
(b) Show that the solution with initial data of the form
u=O,
ut=g(r)
(1.18)
(g=even function of r) is given by
1
u= -2
cr
i
r + ct
pg(p)dp.
(1.19)
for O<r<a
for r>a
(1.20)
r-ct
(c) For
g(r)=
{~
find u explicitly from (1.19) in the different regions bounded by the cones
r=a±ct in xt-space. Show that u is discontinuous at (O,a/c); (due to
focussing of the discontinuity of Ut at t=O, Ixl=a).
2. Consider the initial-value problem (1.1), (1.2) for the wave equation in n=5
dimensions. With Mu(x,r,t) defined by (1.8), set
N (x,r,t)= r2 :r Mu (x,r,t) +3rMu (x,r,t).
(1.21)
(a) Show that N(x,r,t) is a solution of
a
2-aN
-N=c
at 2
ar2
and find N from its initial data in terms of M j and Mg.
2
* For the analogous
2
situation of surface waves in water (n=2) compare Shakespeare (Henry
VI,part I):
Glory is like a circle in the water
Which never ceaseth to enlarge itself
Till by broad spreading it disperse to nought.
109
5 Hyperbolic equations in higher dimensions
(b) Show that
. N(x,r,t)
u(x,t)=lim
3
r .....O
r
3. For X= (Xh X2,X3) consider the equation of elastic waves (see (2.8) of Chapter 1)
LU=(~
-cr~)(~
-d~)U(X,t)=O
at 2
at 2
(1.23)
with positive distinct constants Ch C2.
(a) Show that Mu(x, r, t) defined by (1.8) satisfies
ArM =(~-cr~)(~-ci~)rM =0.
u
at2
ar2
at2
ar2
u
(b) Show that the general solution v(r,t) of Av=O is of the form
v= FI (r+ clt)+ F2(r- clt)+ G I (r+ C2t) + G2(r- c2t).
(c) Solve the general initial-value problem for (1.23) using (a) and (b).
(b) Hadamard's method of descent
In this method solutions of a partial differential equation are obtained by
considering them as special solutions of another equation which involves
more independent variables, and can be solved. For example a solution
U(x\,X2,t) of (1.1), (1.2) with n=2, can be looked at as a solution of the
same problem with n = 3 which happens not to depend on X3. Then
U(X\,X2,t) is given by formula (1.14) for x 3=0 with
g(y) = g(y\,Y2),
f(y) = f(y\,Y2),
the surface integrals being extended over the sphere
Iy - xl =V(y\-x\?+(Y2-X2)2+y~
Observing that on that sphere
110
=ct.
1 The Wave Equation in n-dimensional Space
where
r=y(x l -yl+(X2-Y2i
(1.24b)
We observe that here the domain of dependence of the point (XI,X2,t) on
the initial data consists of the solid disk r<. cl in the YIY2-plane. Thus
Huygens's principle in the strong form does not hold for the wave equation
in two dimensions. Disturbances will continue indefinitely, as exhibited by
water waves.
The same method can be applied to other lower-dimensional equations
as well. Consider, for example, a solution u of (1.1), (1.2) with n = 3 of the
special form
U(X I,X2,X3' I) = eiAx3v(xI,X2' I).
Then v is a solution of the 2-dimensional equation
(1.25)
The solution v of (1.25) with initial values
vt =I/I(X I,X2) for 1=0
V = </>(X I,X2),
(1.26)
is obtained from formula (1.14) for X3 =0, taking
PROBLEMS
1. Write out the solution of the initial-value problem (1.25), (1.26).
2. Show that the solution w(x), t) of the initial-value problem for the telegraph
equation
(1.27)
(1.28)
is given by
(1.29)
Here
s= C2r 2 -(XI-y)2
(1.30)
while J o denotes the Bessel function defined by
Jo(z) =
~ ('1T/2cos(zsinfJ)dfJ.
'lI' Jo
(1.31)
[Hint: "Descend" to (1.27) from the two-dimensional wave equation satisfied by
U(X),X2, t) = COS(Ax2)W(X), t).
Use formulas (1.24a, b).]
111
5 Hyperbolic equations in higher dimensions
3. Solve the initial-value problem (1.1), (1.2) for n=4 by descent from the solution
(1.22) for n = 5.
(C) Duhamel's principle and the general Cauchy problem
Consider the inhomogeneous wave equation
Du(x,t)=w(x,t)
(1.32)
for a function u(x,t) with initial values
u(x,O)= f(x),
ut(x,O)=g(x).
(1.33)
Duhamel's principle permits reduction of the problem (1.32), (1.33) to a
succession of problems of the type (1.1), (1.2) for the homogeneous wave
equation. (The method, the analogue of the method of "variation of
parameters" for ordinary differential equations, applies to more general
linear partial equations.) It is sufficient to consider the problem (1.32),
(1.33) for the special case where the initial data are
u(x,O)=ut(x,O)=O.
(1.34)
We only have to subtract from U the solution of the problem (1.1), (1.2)
which we assume to be known. We claim that the solution of (1.32), (1.34)
is given by
u(x,t)= fotU(x,t,s)ds,
(1.35)
where U(x, t,s) for each fixed s >0 is the solution of
DU(x,t,s)=O
fort>s
(1.36)
with initial data prescribed on the plane t = s:
(1.37)
Ut (x,s,s)= w(x,s).
Let indeed U(x,t,s) be a solution of (1.36), (1.37) of class C 2 in its
arguments for x ERn, 0.;;;; s';;;; t. Then for the u given by (1.35)
U(x,s,s)=O,
ut = U(x,t,t)+ fot Ut (x,t,s)ds = fotUt(x,t,s)ds
utt =
(1.38)
~(x,t,t)+ fot~t(x,t,s)ds
= w(x,t) + fotc2Llx U(x, t,s)ds = w(x,t) + c 2Llxu(x,t),
confirming (1.32). That U satisfies the initial conditions (1.34) is clear from
(1.35), (1.38). Since the wave operator 0 is invariant under translations, we
have in
V(x,t,s)= U(x,t+s,s)
(1.39a)
a solution of the wave equation
DV(x,t,s)=O
112
fort>O
(1.39b)
1 The Wave Equation in n-dimensional Space
with initial values prescribed for 1=0:
V(x,O,s)=O,
~(x,O,s)=w(x,s).
(1.39c)
For n = 3 formula (1.14) yields in
V(x,l,s)=
~I
w(y,s)dSy
4'1TC t Iy-xl=ct
a C 2-solution of (1.39b, c) provided w(x, I) E C 2 for I;> 0 and all x.
Substituting into (1.35) leads to the expression
U(X,/)= fotV(X,I-S,s)ds
=
~ rt ~I
4'1TC )0 I-s Iy-xl=c(t-s)
w(y,s)dSy •
(1.40)
Thus the value of the solution u of (1.32), (1.34) at the point (x, I) depends
only on the values of w in points (y,s) with
ly-xl=c(/-s),
O<s<t,
(1.41)
that is, on the values of w in points of the upper half space lying on the
backward characteristic cone with vertex (x, I). This truncated cone represents the domain of dependence of u(x, I) on w.
The Cauchy problem for (1.32) (and similarly for other equations) with
an arbitrary initial surface S in xl-space can be reduced to that for t=O.
Let the hypersurface S be given by
I = </>(x) =</>(X\,X2,X3)'
We prescribe Cauchy data
u= f(x),
(1.42)
ut=g(x) for (x,/) on S.
Of course, S will have to be noncharacteristic, that is,
\fI(x) = 1- c2~ </>x,';60
(1.43)
(1.44)
i
(see (2.28a, b) of Chapter 3). To simplify the expression we make the
(unessential) assumption that </>,j,gEc oo (1R3), WEc oo (1R4), and that </»0.
To find a solution u of
Du(x,/)=w(x,/) for I > </>(x)
(1.45)
with Cauchy data (1.43), we first construct an approximale solulion v of
order 2, i.e., a function v(x, I) which just on S satisfies
v=f,
a
a2
Dv-w= -(Dv-w)= -(Dv-w)=O. (1.46)
al
a/2
We can always find a special v satisfying (1.46) in the form of a 4th-degree
polynomial in I:
4
v(x,/)=
L ai(x)(/-</>(x»)i
(1.47)
i=O
113
5
Hyperbolic equations in higher dimensions
(this is just the beginning of the formal Cauchy-Kowalewski expansion for
u in terms of powers of t-q,). The coefficients aj(x) have to be found
recursively from
ao=j,
al=g
21/1a2 + c2(~(alq,- ao) -q,~al) = w(x,q,)
61/1a3+ c2(~(2a2q,- a l )
2q,~a2) = wt(x,q,)
-
241/1a4 + c2(~(6a3q,- 2a2) - 6q,~a3) = wtt(x,q,).
(1.48a)
(1.48b)
(1.48c)
(1.48d)
The aj exist because of (1.44). We change our problem (1.45), (1.43) to one
for the function U = u - v. We then want
OU(x,t)= W(x,t)
for t>q,(x)
U= Ur=O for t=q,(x).
(1.49a)
(1.49b)
Here
W(x,t)=W(x,t)-Ov(x,t)
(1.50)
belongs to C OO(1R4) and, by (1.46), satisfies
W(x,t)= W,(x,t)= w,t(x,t)=O for t=q,(x).
(1.51)
Then all first and second derivatives of W vanish for t=!f>(x) and the
function W* defined by
W(x,t)
W* ( x t ) = (
,
0
for t>q,(x)
fort<;q,(x)
(1.52)
belongs to C 2(1R4). We replace the Cauchy problem (1.49a, b) by the
initial-value problem
OU(x,t)= W*(x,t)
for t>O
U(x,O)= Ur(x,O)=O
(1.53a)
(1.53b)
for which we can find a solution of class C 2 for t > 0 with the help of
formula (l.40). We claim that the U found from (1.53a, b) satisfies (1.49a:
b) provided S is spacelike in the sense that
1/1 (x) = 1- c 2 ~ q,:,> O.
(1.54)
j
Clearly (1.53a) implies (1.49a), since q,>0. Moreover, (1.49b) follows if we
show that U(x,t)=O for t<q,(x). Now by (l.40) U(x,t)=O if W*(y,s)=O
in all points of the backward characteristic cone (see Figure 5.4)
Ix-yl<c(t-s),
O<s<t
(1.55)
with vertex (x,t). By (1.52) this will be the case if we can show that
s<q,(y) whenever (1.55) holds and t<q,(x). Now s>q,(y) would imply
114
1 The Wave Equation in n-dimensional Space
s
t = rj>(x)
t=O
Figure 5.4
that
Ix-yl..;: c(t- s) < c{cp(x) -cp(y)) = c ~ cpx,(n(xj-Yj)
j
<'Y~ cp~(~)
Ix-yl
with an intermediate point r Using (1.54) this leads to a contradiction, and
(1.49b) is established.
In this way the existence of a solution of the Cauchy problem with data
on a spacelike surface S is established. The argument will not work if S is
not spacelike, since then the cone (1.55) might contain points where
W*~O. Data on "timelike" surfaces with l/I(x)<O or surfaces which are
partly spacelike and partly timelike cannot be prescribed arbitrarily, since
then some points of S lie in the domain of influence of others.
Uniqueness for the Cauchy problem (1.45), (1.43) is trivial. Let u be a
solution with homogeneous dataf=g=w=O and of class C 2 for t~cp(x).
We can then continue u as identically 0 to all (x,t) with 0<. t<cp(x). Then
u becomes a C 2-solution of Du = 0 with vanishing initial data for t = 0, and
hence vanishes identically.
115
PROBLEM
Let S denote a spacelike hyperplane with equation t=yx, in xt-space. Show that
the Cauchy problem for Du = 0 with data on S can be reduced to the initial-value
problem for the same equation by introducing new independent variables x',t' by
the Lorentz transformation
(d) Initial-boundary-value problems ("Mixed" problems)
So far we have considered the "pure" initial-value problem for the wave
equation, with x ranging over the whole space ~n for t>O. We next
consider solutions u(x, t) of
for xEn, t>O,
Du=w(x,t)
(1.56)
where n is an open set in x-space. For simplicity we take n=3. We
associate with the operator 0 the energy integral
E(t)=
l!( Ul+C2~ U~)dx
o
(1.57)
I
(see (1.16)*). Then, using (1.56)
~=
=
=
l(
1(C2~
o
o
UtUtt + c 2
~ UxYx,t) dx
I
(utux)x, +UtW)dx
I
rutwdx+ J(aoc2utddun dSx'
)0
(1.58)
where d / dn denotes differentiation in the direction of the exterior normal
to an. If here
U= ut=O for xEn, t=O
u=O
or du/dn=O for xEan, t>O
Du=O for xEn, t>O
(1.59a)
(1.59b)
(1.59c)
it follows that E(t)=O for t>O, since E(O) =0, dE/dt=O. Since the
integrand in (1.57) is a definite quadratic form in the first derivatives of u,
we conclude that ut = ux . = 0 for x En, t > 0, hence that U is constant, and
thus u=O because u=O initially. Consequently (subject to appropriate
regularity assumptions) a solution u of (1.56) is determined uniquely given
* In many applications E
116
represents the sum of kinetic and potential energy.
1 The Wave Equation in n-dimensional Space
(a) the values of w for xEQ, t>O, (b) initial values
u=f(x),
ut=g(x)
forxEQ,t=O,
(1.60)
and (c) boundary values ("Dirichlet data")
for xE aQ, t>O
u=h
(1.61)
(where instead of u, du / dn can also be prescribed on aQ).
In the special case where w = h = 0, the solution can be found by
expansion into eigenfunctions for the Laplace operator for the region Q, in
analogy to (4.21) of Chapter 2. An eigenfunction v(x) corresponding to the
eigenvalue A is a solution of
dV(X)+AV(X)=O for xEQ
(1.62a)
v (x) = 0 for x E aQ,
(1.62b)
where v does not vanish identically. Under appropriate regularity assumptions on Q there exists a sequence of eigenvalues Ak and a corresponding
sequence of eigenfunctions vk(x) which form a complete orthonormal set
on n. (See Courant-Hilbert [7].) This leads to an expansion
u(x,t)= ~ak(t)vk(t)
(1.63)
k
for the solution u(x, t) of our initial-value problem. Substituting into the
equation u=O and comparing coefficients, one finds that the ak(t) are
solutions of the ordinary differential equation
ak(t)+c2~ak(t)=0.
(1.64)
Using the initial conditions (1.60) and Fourier's formula
ak =
k
UVk dx
(1.65)
for the coefficients of an expansion in terms of orthonormal functions, one
arrives at initial conditions
(1.66)
Equations (1.64), (1.66) easily permit us to determine the ak(t) as trigonometric functions of t:
a,(t)- [ [ J(X)COs(cl.,t) +
g(X):~cl.,t) ]V,(X)dx.
(1.67)
Some mixed problems can be solved "in closed form" when the space
region Q is the halfspace Xn > O. The principal tool here is the extension to
the whole space by reflection. Let u be the solution of
Du= w(x,t)
u=f(x),
for X3 >0, t >0
(1.68a)
ut=g(x)
(1.68b)
U=h(XI,X2,t)
forx3>0,t=0
for X3=0, t>O.
(1.68c)
117
5 Hyperbolic equations in higher dimensions
Relations (1.68a, b, c) imply certain consistency conditions between the data
j,g,h, W, at least if a solution u of class C 3 is to exist in the closed quadrant
X3~0, t~O:
j=h,
c 2/lg + W t = hili
g=h"
for X3 = t = O.
(1.69)
We shall construct a solution u of class C 2 for X3 ~ 0, t ~ 0, provided the
dataj,g,h, w, have sufficiently many derivatives in their domains of definition and satisfy (1.69). Moreover the solution is unique.
For that purpose we first simplify the problem by making use of an
approximate solution v(x,t) oj order 2 of (1.68a, b) (see p. 113), that is, a
solution of
Oi
- . (Ov- w)=O for i=O, 1,2 when x 3=0, t ~O. (1.70)
v=h,
OX3
We can always find such a v that is a polynomial in X3
4
V=h(Xl,X2,t)+ ~ ai(Xl,x2,t)X~
(1.71)
i=2
with suitable coefficients a2,a3,a4' It remains to find a solution U(x,t)=
u(x,t)-v(x,t) of
oU= W= W -
Ov for X3 ~ 0, t ~ 0
U= F(x) = j(x) - v(x,O),
Ut = G (x) = g(x) -vt(x,O)
for X3 ~ 0, t=O
U=H=O for x 3=0,
(1.72a)
t~O.
(1.72b)
(1.72c)
Observe that here by (1.70), (1.69)
W = WXjX3 = 0 for X3 = 0, t ~ 0
F=FX3X3 =G=GX3X 3 =0 forx3=0.
(1.73a)
(1.73b)
If we extend F, G, W by "reflection" to all values of X3' so as to be odd in
X3' then the resulting functions F*, G*, W* will be defined for all x.
Moreover by (1.73a, b) F* E C 3 , G* E C 2 for all x, and W* E C 2 for all x
and t ~ O. We can then solve the pure initial-value problem
U= W*
U=F*,
for t>O
(fr= G* for t=O
using formula (1.40) to obtain a C 2-solution. Obviously the restriction of U
to values X3 ~ 0, t ~ 0 satisfies (1.72a, b). Moreover U is odd in x 3, since
U(Xl,X2,X3,t)+ U(Xl,X2' - X3,t)
118
I The Wave Equation in n-dimensional Space
will be a solution of the wave equation with initial values 0, thus vanish
identically. This also implies that (1.72c) holds. Finally a C 2-solution u of
(1.68a, b, c) is unique. Since for j=g=h=w=O it could be extended by
reflection to a solution of the wave equation with initial values that are 0
everywhere, and hence would vanish.
We observe that the consistency conditions (1.69) are satisfied automatically, when h =0 and in addition j,g, w vanish for all sufficiently small x 3•
PROBLEMS
1. Let n denote an open bounded set in n-dimensional x-space described by an
inequality ![>(x) >0, so that q,(x)=O on an. Let SA for A;;;' 0 denote the hypersurface in xt-space given by t=Aq,(X) for x En. On SA define for a function
u(x,t) the energy integral
E(A)=
where
QA =
i (u? +
c2
1. QA dx,
(1.74a)
s~
~ U;' ) + Ac2ut ~ UXiq,x/
(1.74b)
(a) Prove E(A)=const. when Ou=O. [Hint: Integrate utOu over the lens-shaped
region O<t<Aq,(X).]
(b) Show that QA as a quadratic form in u"ux" ... ,uxn is positive definite, when
SA is spacelike.
(c) Show that the initial data on So of a solution of Ou=O uniquely determine u
on all SA with sufficiently small A. (Compare Holmgren's theorem, p. 66.)
2. Let u be a solution of (1.68a, b, c) where f = h = w = O. Find the domain of
dependence of u(x,t) on g.
3. Consider the mixed problem for u(x,t)=U(XI>X2,x3,t)
(1.75a)
Ou=O for X3 >0, t>O
u=f(x),
u,=g(x)
forx3>0,t=0
Mu=O for X3=0, t>O,
(1.75b)
(1.75c)
where M denotes a first-order operator of the form
(1.75d)
with constant coefficients a;, and f,g vanish for all sufficiently small X3 >0.
Prove there exists a solution u provided a3";; O. (Compare problem 2, p. 43.)
[Hint: First determine v = Mu for X3 > 0, t > 0 from its initial and boundary
conditions as a solution of Ov = O. Next find u for X3 > 0, t > 0 as a solution of
Mu = v with initial condition u = f by the methods of Chapter 1. Verify that the
u obtained satisfies (1.75a, b, c).]
119
2. Higher-order Hyperbolic Equations
with Constant Coefficients
(a) Standard form of the initial-value problem
For functions u(x,t)= U(XI, ... ,Xn,t) we define the differentiation operators
D=(DI, ... ,Dn)=(
a: a:
1
, ... ,
n
)'
r= :t
(2.1)
where D is the gradient vector with respect to the space variables. Using
the Schwartz notation of Chapter 3 we can write the most general mthorder linear partial differential equation with constant coefficients in the
form
(2.2)
P(D,r)u=w(x,t),
where P(D,r)=P(DI, ... ,Dn,r) is a polynomial of degree m in its n+ 1
arguments. We associate with equation (2.2) in the half space t>O the
initial conditions
rku =lk(x)
fork=O, ... ,m-l andt=O.
(2.3)
We shall assume that the plane t=O is noncharacteristic. This means that
the coefficient P(O, 1) of rm in the polynomial P does not vanish. Dividing
by a suitable constant we can bring about that
P(O,I)=l.
(2.4)
Problem (2.2), (2.3) for general data W,jk can be reduced to the standard
problem where the data have the special form
w=10= II = ... =Im-2=0,
1m-I =g(x).
(2.5)
The solution of the standard problem (unique by Holmgren's theorem) will
be denoted by ug(x, t). To achieve this reduction we first find a solution u
of (2.2) with zero initial data. Such a solution is furnished according to
Duhamel's principle by the formula
u(x,t)= fotU(x,t,s)ds,
(2.6)
where U (x, t, s) for each parameter value s > 0 is the solution of the
initial-value problem
P(D,r)U(x,t,s)=O
fort;;;.s
rkU(x,t,s)=O for k=0, ... ,m-2 and t=s
rm-IU(x,t,s)= w(x,s)
for t=s.
(2.7a)
(2.7b)
(2.7c)
That it solves (2.2) is easily verified, using (2.4). Here for each s ;;;. 0 the
120
2 Higher-order Hyperbolic Equations with Constant Coefficients
function U(x,t,s) is found by solving a standard problem; in fact
U(x,t,s)=ug(x,t-s) where g(x)=w(x,s).
(2.8)
It remains to reduce the solution of the homogeneous equation
P(D,T)U=O
(2.9)
with general initial conditions (2.3) to standard problems. For that purpose
we arrange the polynomial P(D,T) according to powers of T:
P(D,T)=Tm+PI(D)T m- I+ ... +Pm(D),
(2.10)
where Pk(D) is a polynomial of degree <.. k in DI, ... ,Dn. Using the
differential equation (2.9) one easily verifies that the solution U with initial
data (2.3) is representable in terms of the standard problems associated
with each individual A by the formula
u= ufm_l +( T+ PI (D »)ufm_2 +( T2+ PI (D )T+ PiD »)ufm _3
+ ...
+(Tm-I+PI(D)Tm-2+PiD)Tm-3+ ... +Pm-I(D»ufo . (2.11)
As an example we have for the solution of the wave equation
(T2 - c2,:l)u = 0
with initial values
U=j,
u/=g fort=O
the formula
in agreement with (1.14).
A system of N linear partial differential equations of order m for N
functions UI, ... , UN can also be written in the form (2.2), where now U
stands for the column vector with components UI' ... 'UN, and P(D,T) is a
square N X N matrix whose elements are polynomials of degree <.. m in
DI, ... ,Dn,T. The data w,A in (2.2), (2.3) are column vectors. The solution
ug of a standard problem corresponds to the data (2.5). For a noncharacteristic initial plane t = 0 the matrix P (0, I) is nondegenerate, and we can
assume that
P(O,I)=1
(2.12)
is the unit matrix. The solution of (2.2) with zero initial data still is
described by Duhamel's formulas (2.6), (2.7a, b, c), and thus reduced to
standard problems as in (2.8). The reduction of general initial data to
standard ones is achieved by a modification of (2.11) which reads
m-2up .+ ... +up .).
u=uj,m-i + (TUj,m-2 +uPj,
)+ ... +(T m- IU.+T
I m-2
JO
)JO
m-lJO
(2.13)
121
5 Hyperbolic equations in higher dimensions
In what follows we shall only have to deal with the standard problem
P(D,r)u=O
'l"ku=O
fort::>O
for k=0, ... ,m-2 and t=O
'l"m-lu=g(x)
for t=O.
(2.14a)
(2.14b)
(2.14c)
We call the differential equation or system of equations (2.14a) hyperbolic (with respect to the plane t=O), if the initial-value problem (2.14a, b,
c) has a solution u(x,t) of class em, for all g(x) E CO (Rn), where s is
sufficiently large. * We also say that the plane t = 0 is spacelike.
PROBLEM
Verify that formulas (2.11), respectively (2.13), give the solution of the initial-value
problem (2.3), (2.9).
(b) Solution by Fourier transformation
Following Cauchy a formal solution of the standard problem (2.14a, b, c)
can be obtained by Fourier transformation with respect to the space
variables. It will be an actual solution if the integrals involved converge
adequately. We associate with a function g(x)E CO (Rn) its Fourier transform g, defined by
(2.15)
(x·~= Xl~l + ...
formula
+ xn~n).t For g E CO with sufficiently large s the reciprocal
(2.16)
holds. We find from (2.15) by integration by parts for any k=I,2, ... ,n
that
f Dk(e-iX'~) g(x)dx
= _(2'1T)-n/2 f e-ix·~Dkg(x)dx.
- i~k,g(~) = (2'1T)-n/2
We write this fundamental identity as
./::
A
~
lSkg=Dkg·
(2.17)
* Using the finiteness of the domain of dependence of u on g (implied, e.g., by Holmgren's
theorem), one can show that in the hyperbolic case the problem (2.14a, b, c) has a solution for
gE C'(Rn), even without the assumption of compact support.
t Here, of course, i = v=t . Observe that generally g is complex valued, even when the
variables x,~,g are restricted to real values. In what follows the independent variables x,~ will
be assumed to be real, unless the contrary is stated, but g,g, and the coefficients of the
polynomial P will be allowed to be complex valued.
122
2 Higher-order Hyperbolic Equations with Constant Coefficients
By repeated application we find more generally for g E
multi-index a=(al, ... ,an ) with lal";; s that
CO
and any
(i~)ag=t?g.
(2.18)
Thus differentiation for g is transformed into multiplication for g.
Formula (2.18) permits us to show that g(~ decreases rapidly for
when s is large. Let ~ = (~l' ... '~n)' where, say,
~--700
(2.19a)
Then
1~I=y~~f
..;; 2sn s / 2
L
";;Vn
lal';;s
I~I
(2.19b)
(2.19c)
I~al·
Consequently
(1 + I~IYI g (~)I..;; 2sn s / 2
..;;2sn s/ 2
L
lal<s
l(i~tg (~)I
L !IDag(x)ldx";; Ms
<00,
(2.20)
lal.;;s
where Ms depends on n,s, the size of the support of g, and the maxima of
the absolute values of the derivatives of g of orders ..;; s. It follows in
particular that
forgECn+1(lRn)
IgA(~)I~ Mn+l
"" (I+IW n + 1
(2.21)
and hence that the integral in (2.16) converges absolutely. Formula (2.16)
is valid for s> n.
Let now u(x,t) be a solution of (2.14a, b, c). To begin with we work
with a single partial differential equation, so that u is a scalar. We write
tentatively
f
u(x,t)=(2'1T)-n/2 eixo€fi (~,t)d~,
(2.22)
where u(x, t) is the Fourier transform of u with respect to x. Purely
formally we obtain by differentiation
f
0= p (D,'T)u(x, t) = (2'1T) -n/2 eixol;p (i~,'T)u (~,t)d~.
123
5
Hyperbolic equations in higher dimensions
In addition for t =
°
for k=O, ... ,m-2
f
-{ :(X)=(2W)-"/2 e"'i(g)dE
for k=m-l.
These relations are satisfied formally, when
solution of the ordinary differential equation
u(~,t)
for each
~ElRn
is a
(2.23a)
P(i~,'T)U (~,t)=O
with initial values for t = 0
for k=O, ... ,m-2
for k=m-l.
(2.23b)
This leads to the formal solution* of (2.14a, b, c)
(2.24a)
where Z as a function of t denotes the solution of the ordinary differential
equation problem
P (i~,'T)Z (~, t) = 0
(2.24b)
with initial values for t = 0
for k=0, ... ,m-2
for k=m-l.
(2.24c)
There is no problem with the existence of Z. Moreover we can verify
directly that the u given by (2.24a) is of class em in x, t for x ElRn, t ~ 0,
and actually satisfies (2.14a, b, c), if g E q with s> n, and g and Z are
such that all differentiations with respect to x or t of orders ..; m can be
carried out under the integral sign in (2.24a). This is certainly the ~ase
when the resulting integrals converge absolutely. For that it is sufficient
that the expressions
(1+I~Dn+II'Tk~az(~,t)g(~)1
forlal+k..;m
(2.25)
are bounded uniformly in ~, t for all ~ E IRn and for t restricted to any finite
interval 0..; t"; T.
Of course, the expressions (2.25) will be bounded in any bounded set in
~t-space. What matters is only the behavior for large I~I. Here, to a certain
extent, g(~ can be controlled by assuming that s is large enough, as is
*More precisely our arguments show that it there exists a solution u(x,t) of (2.14a, b, c) of
compact support in x and sufficiently often differentiable, then u must be given by the
expression (2.24a).
124
2 Higher-order Hyperbolic Equations with Constant Coefficients.
shown by the estimate (2.20). It is just a question of the growth of Z (~, t)
and its I-derivatives. If we can show that there exists a constant N, such
that
we find for the expressions (2.25), using (2.20), the upper bound
(I +1~lr+l+m-sNMs.
For the boundedness of the expressions (2.25) it is here sufficient to
assume that
s> n+ I+m.
(2.27)
Formula (2.24a) will then represent a solution of our standard Cauchy
problem (2.l4a, b, c).
The proper condition on the partial differential equation (2.l4a), i.e., on
the polynomial P, under which an estimate of the form (2.26) holds, and
hence the initial-value problem can be solved, is:
Girding's hyperbolicity condition. Equation (2.l4a) is hyperbolic if there
exists a real number c such that
P (i~, iX) ~O for all ~ ERn and all complex X with ImX, - c. (2.28)
Condition (2.28) is equivalent to the statement that all of the m roots X of
P (i~,iX)=O
(2.29)
lie in one and the same half plane
ImX> -c
(2.30)
of the complex number plane for all real vectors ~. *
To establish the sufficiency of Garding's condition we represent the
solution Z of (2.24b, c) by a Cauchy integral:
Z
(~,
l
iAr
1
e
dX
) - 2'17" r P (i~, iX) ,
I -
(2.31)
where the closed path of integration r runs around each root X of (2.29)
once in the counterclockwise direction. Indeed differentiation of Z as
defined by (2.31) with respect to t results in mUltiplying the integrand by iX
• Girding showed that his condition (2.28) is necessary as well as sufficient. An even stronger
statement holds in the case where the polynomial P(D,T) is irreducible (i.e., not representable
as product of lower degree polynomials): If the equation (2.14a) is not hyperbolic, the
initial-value problem (2.14a, b, c) for gEq(Rn) never has a solution, unless g vanishes
identically.
125
5
Hyperbolic equations in higher dimensions
so that
P{i~'T)Z=-21'TT
1
P{i~,{A) P
r
t")
"N
1~,IA
dA
= _I r eiNdA=O
2'TT Jr
by Cauchy's theorem, while for t=O by (2.10)
TkZ=_I- (
ikAk
dA.
2'TT r imAm+Pl{i~)im-lAm-l+ ... +Pm{i~)
J
Expanding r to infinity we see that this expression has the value 0 for
k=0, ... ,m-2, and the value I for k= m-l.
We first derive an upper bound for the roots A of (2.29). Using the
expansion (2.10) we have
P{itiA)={iA)m+PI{i~){iA)m-l+ ... +Pm{i~)=O
(2.32)
For each of the kth-degree polynomial Pk we have a trivial estimate
IPk{i~)I' M{I + 1~ll for all ~ElIln
with a suitable constant M. Then for a root A of (2.29)
(2.33)
m
1i\lm, M ~ (I + IwkIAlm-k.
k=l
Setting 0= IAI/(I + I~I), we have then
om'M{1+0+02+ ... +om-I).
This implies that either 0 < I or om, MmO m- 1 and hence 10 I< Mm. Thus
for the roots A of (2.29)
IAI
0= 1 + I~I
< 1+ Mm.
(2.34)
Denote by Ak(~) for k= 1, ... ,m the m (not necessarily distinct) roots A
of (2.29) taken in any order. Then
m
P{i~,iA)=im
II
(A-i\k{~»)'
(2.35)
k=l
Take for each k = 1, ... , m the open disk of center Ak and radius 1 in the
complex A-plane. Let U denote the union of these m (possibly overlapping)
disks. Take for the path of integration r in (2.31) the boundary of U,
which possibly consists of several closed curves and is composed of pieces
of the boundaries of the individual unit disks. Then r runs once around
each of the ~ and has total length ,2m'TT. Moreover each of the points A
of r has distance # 1 from each of the Ak, so that by (2.35)
IP{i~,iA)1 # 1
126
for AEr.
2 Higher-order Hyperbolic Equations with Constant Coefficients
Since each point of f has distance 1 from some Ak we have from (2.34) and
the Garding condition (2.30)
ImA;> -c-I,
IAI ~ 1 +(1 + Mm)(I +I~I) ~(2+Mm)(1 +I~I) on f.
Thus
fort;>O,AEf.
leiAtI~e(l+c)t
f
It follows from (2.31) that
1
IrkZ(~,t)l= 2",
~
(iA)ke iAt
P(i~,iA) dA
m(2+ Mm)k(1 + IWke(l+C)T
(2.35a)
for 0 ~ t ~ T, ~ E IR n , 0 ~ k ~ m. This is an estimate of the type (2.26). It
follows that for gE CO·+m+I(lRn) the initial-value problem has a solution of
class em for t;> 0, provided the Garding condition (2.28) is satisfied.
The integral (2.31) for Z is easily evaluated by the calculus of residues,
in the case where all the roots Ak are distinct. One finds that then
m
.
e tAkt"A
Z(~,t)= '"
£.J P C~
k=1
T
1 ,1 k
r
(2.36)
As an example consider the n-dimensional wave equation corresponding to
the operator
n
P(D,r)=O=r2-c 2 ~ Df.
(2.37a)
k=!
Here
(2.37b)
has the real roots
(2.37c)
satisfying the Garding condition. Then by (2.35)
Z (~, t) =
sin(cl~lt)
cl~1
(2.37d)
Thus the standard problem for the wave equation has the solution
(2.37e)
for gE eO+ 3(lRn).
If the polynomial P(D,r) is homogeneous of degree minD and r (as in
equation (2.37a», we have for every solution (~,A) of (2.29) and every real s
P(S~,SA)=O,
Im(sA)=slmA.
127
5
Hyperbolic equations in higher dimensions
Here sImA can be bounded from below for all s only, if ImA=O. Thus the
Garding condition for homogeneous P is that all roots A of the equation
P (i~,rA) = imp (~,A) =0
(2.38)
are real for all real ~.
In many cases hyperbolicity can be inferred from properties of the
principal part of P alone. We arrange the terms in the polynomial according to their degree, writing
P(D,r) =Pm(D,T)+Pm_I(D,T) + ... +PO(D,T),
(2.39)
where Pk(D,T) is a form of degree kinD and T. HerePm(D,T) is identical
with the principal part of P(D,T) as defined in Chapter 3. We shall prove:
For the Garding condition for P to be satisfied it is necessary that all
roots A of
(2.40)
are real for all real ~ (i.e., that Pm satisfies the condition); a
sufficient requirement which implies the Garding condition for P is
that all roots A of (2.40) are real and distinct for all real ~+O.
To prove this statement we apply the substitution
~= PT/,
(2.41)
A= pp.,
where p= I~I and T/ is a unit vector. Then P(i~,iA)=O goes over into the
equation
1
Pm (T/,p.) + ip Pm-I(T/,P.) + ...
1
+ {ip)m Po(T/,p.) =0
(2.42)
for p., depending on the parameters p, T/. By (2.4) the coefficient of p. m in
(2.42) has the value 1. The coefficients of the powers of p. not contributed
by the principal part tend to 0 for P-'HXJ, since T/ is bounded. Using the fact
that the roots of a polynomial with highest coefficient 1 depend continuously on the coefficients, we see that for p~oo the roots p. of (2.42) will
tend * to the roots of
(2.43)
Let there exist for a certain T/ a root ILo of (2.43) with ImlLo+O. Assume
1m ILo = - y < 0 (otherwise replace T/ by - T/ and ILo by - /Lo). Then there
exist roots p. of (2.42) for all sufficiently large p for which Imp. < - y /2 and
hence roots A of (2.29) for which ImA < - py. This contradicts (2.30) for
large p. Thus necessary for (2.30) is that the roots p. of (2.43) are real for all
real'tJ with IT/I = 1, and then also for all real T/. Assume next that the roots p.
are real and distinct for real T/+O, in particular for IT/I = 1. We now use the
* More precisely in a given neighborhood of a root p. of (2.43) of multiplicity y there lie
precisely y roots of (2.42) if p is sufficiently large.
128
2 Higher-order Hyperbolic Equations with Constant Coefficients
fact that roots of a polynomial equation with highest coefficient one are
differentiable (even analytic) functions of the coefficients in any region not
containing multiple roots. They will be uniformly Lipschitz continuous in
any compact subregion. For large p and 11J1 = 1 the coefficients of equation
(2.42) for p. differ from those of equation (2.43) by terms of order 1/p.
Hence the difference of the roots p. of (2.42) from appropriate roots of
(2.43) is of order 1/p uniformly for 11J1 = 1. Since (2.43) has real roots, it
follows that the imaginary parts of the roots p. of (2.42) are of order 1/p,
and hence the imaginary parts of the roots A. of (2.29) are bounded
uniformly for all sufficiently large p= I~I. By (2.34) A. and ImA. also are
bounded for bounded I~I. Thus (2.30) follows.
We call P strictly hyperbolic when its principal partpm(~'A.)=O has real
distinct roots for ~~O hyperbolic. We see that strict hyperbolicity implies
~yperbolicity. Thus, for example, any equation of the form
(2.44)
uIt =c 2 b.+ku
is hyperbolic.
Formula (2.24) for the solution u(x, t) of the standard initial-value
problem makes use of the values of g(~, which by (2.15) depend on the
values of the given function g at all points. Actually by Holmgren's
theorem the domain of dependence of u(x,t) on the values of g is known
to be finite; equivalently initial data g of compact support lead to solutions
u(x,t) of compact support in x. This is not obvious from the expression
(2.24a), but can be deduced for strictly hyperbolic P from a version of the
Pailey- Wiener theorem. This involves a shift in the integrations in (2.24a)
to complex ~. From this we require estimates for the functions Z and g for
complex arguments ~ + ir and real t >0, where ~ and r are real.
Assume that the function g(x) belongs to CO (II~n) where s> n + m + 1,
and that the support of g(x) lies in a balllxl < a. For the complex vector
~+ ir we define I~+ irl by
I~ + ir 12 =
n
L I~k + irk 12 = 1~12 + Ir 12.
(2.45)
k=i
Then as in (2.I9c) and by the same arguments
(1+I~+irIY<2sns/2
L
lal<s
I(~+irtl
(2.46)
We conclude from (2.15), (2.18) in analogy to (2.20) that
(1+I~+irIYlg(~+inl<2sns/2
Lf
lal<s Ixl<a
<2sns/2ealrl
le-jx·(~+j.nDag(x)ldx
L JIDag(x)ldx<ealrIMs'
(2.47)
lal<s
129
5
Hyperbolic equations in higher dimensions
since for real
x with Ixl < a
Ie -ix.(€+iol
= e x·r OS;;; elxllrlos;;; eo1rl.
We proceed to estimate
Z{~+·S t)- 27TI
I,
-
1
r
e iN
P(i{~+ in,iA)
(2.48)
dA
using as path of integration r again the boundary of the union of the unit
disks with centers at the roots Ak of
P(i{~+in,iA)=O.
(2.49)
It follows, as in (2.35a) for k=O that
IZ{~+is,t)los;;; me(l+c)t,
(2.50a)
where
(2.50b)
To estimate c we apply the substitution
~+
is = fY/1,
A= PJL,
(2.51)
where p= I~+ i1J1 and 1J is a complex vector with 11J1 = 1. For a root A of
(2.49) we obtain again equation (2.42) for JL. The coefficients in equation
(2.42) differ from those in the equation
Pm (1J,JL) =0
(2.52)
by terms of order I/p. Since 1J=(~+in/p, and 1~/pl,ls/pl<1 the
coefficients in the equation (2.52) differ from those in the equation
Pm(~/P,JL)=O
(2.53)
by terms of order Isl/p. Since the roots JL of (2.53) are real, it follows for
the roots JL of (2.42) that
ImJL=O( I :Isl).
Thus the roots A of (2.49) satisfy
ImA=O(I+lsl)·
(2.54)
Since also as in (2.34).
IImAlos;;; IAI =0(1 +I~+ is/)=O(I +p)
is bounded for bounded p, we see that (2.54) is valid for all ~+ is. Thus
there exists a constant M such that for the roots A= Ak of (2.49)
130
2 Higher-order Hyperbolic Equations with Constant Coefficients
and hence, using (2.50a), (2.47)
IZ(~+
in,t)1 ~ me(l+M+Mlfl t)
leix-(~+inZ (~+ ir, t) g (~+
inl ~
mM e-x·f+t+Mt+(a+Mt)lfl
s
(1+1~IY
(2.55)
By Cauchy's theorem we can in (2.24a) shift the domain of integration
from that of real to ~ + ir with fixed r without changing the value of the
integral, due to the decay of the integrand for large I~I. Choose now r to be
of the form ux/lxl, where 17>0. It follows from (2.24a), (2.55) that
f
lu(x,t)1 ~(2'1T)-n/2mMset+Mt-(J(lxl-a-Mt) (1 + I~I)-S d~.
If here
Ixl>a+Mt
it follows for u~oo that u(x,t)=O. Hence u for each 1>0 has bounded
support lying in the ball Ixl ~ a + Mt. The constant M here represents an
upper bound for the speed of propagation of disturbances.
So far we have dealt with the standard problem for a single scalar
equation P(D,r)u=O. The case of a system of equations with constant
coefficients requires only minor adjustments. If P is an N X N matrix
satisfying (2.12) a formal solution of (2.14a, b, c) is again furnished by
(2.24a), where now, however, Z (~, t) is an N X N matrix given by
Z
(~, t) = 2~ Irei>.t {p (i~, iA)-1 dA.
(2.56)
Here r has to be a path in the A-plane enclosing all singularities of the
matrix P - I, that is, all of the mN roots Ak of the equation
Q(i~,iA)=detP(i~,iA)=O.
(2.57)
Garding's hyperbolicity condition for systems states that there exists a
constant c such that
ImA> -c
for all roots A of (2.57) for all real vectors ~.
J. Hadamard introduced the important distinction between well-posed
(also called correctly-set) problems and those that are ill posed (improperly
posed, incorrectly set). The distinction applies specially to problems where a
"solution" u is to be found from "data" g. Well-posed problems are those
for which
(a) u exists for "arbitrary" g.
(b) u is determined "uniquely" by g.
(c) u depends "continuously" on g.
Here the words in quotation marks are somewhat vague and require that
131
5
Hyperbolic equations in higher dimensions
the spaces of admitted functions u .and of functions g are specified.
Typically well-posed problems are the Dirichlet problem for the Laplace
equation and the initial-value problem for a hyperbolic equation with
constant coefficients. * (see [27]).
The initial-value problem for the scalar equation P(D,'T)u=O certainly
is ill-posed when the principal part Pm of P does not satisfy Garding's
condition, that is, when there exists a real vector 'I] and a nonreal scalar ILo
such that
We can assume here that
ImlLo= - y<O,
1'1]1= 1.
Consider for any s exponential solutions of the form
(1 + I~I) -s-mei(xoHAt),
where
P(i~,iA)=O
(2.58)
and s is an arbitrary integer. Take here
A=P/L·
For sufficiently large P we can find a A for which I/L -1Lo1 < y /2. Then
~=p'l],
IAI «11Lo1 + h )p,
so that for t =0 and lal < s, 0 < k < in
ImA<
IDa.r"u1 = (I + p) -s-mIAlkl~al
-hp,
t
«1 +p)-s-m(1 /Lol + h pk+la l
«1+1 /Lo1+ ht,
while
u(O,t) = (I-I:p) -s-mleiAtI >(1 + p) -s-me ypt/2.
Thus the initial data and their derivatives of orders < s are bounded
l.miformly for all x, while u(Q,t)~oo for p~oo and any fixed t>O. Here u
does not depend "continuously" on its initial data.
PROBLEMS
1. For n=3 identify the solution of the standard initial-value problem for the wave
equation given by (2.37e) with the solution u= tMg(x,ct) obtained from (1.14).
[Hint: Compute Mg(x,ct) in terms of g from (2.l6).]
*In the latter problem one can specify that u E em for x ERn, t;;. 0 and that g has uniformly
_bounded derivatives of orders .;; s with s chosen sufficiently large. Then u depends continuously on g in the sense that the maximum of lui can be estimated in terms of the maxima of
the IDQgl for lal.;; s. More generally the Cauchy problem with data on a space like surface is
well posed.
132
2 Higher-order Hyperbolic Equations with Constant Coefficients
2. Solve the standard initial-value problem for the system of equations of elastic
waves
(2.59)
(with positive constants p,A,JL) in the form (2.24a), computing the matrix
explicitly from (2.56). [Answer: Z (~, t) is the matrix with elements
Cl( ~ikl~12_~i~)Sin(C21~lt) + c2~~kSin(Cll~lt)
Z(~,t)
(2.60)
where cr=(A+2JL)/p, ci=JL/p.]
3. Show that for n = 1, m = 1 and any N the system of equations
Ut + Bux - Cu = 0
is strictly hyperbolic, when the matrix B has real and distinct eigenvalues.
(Compare (5.12) of Chapter 2.)
4. Prove that, when Girding's condition is satisfied, the solution u of (2.l4a, b, c)
can be written
J
u(x,t)=(1-~xY K(x-y,t)g(y)dy,
(2.61a)
where
(2.61b)
and s is any integer exceeding n/2. [Hint: Introducing h(x)=(l-~xJ'g(x), we
can substitute for g in (2.24a) the expression (1 + 1~12)-Sh(~. Interchanging the
integrations yields
u(x,t)=
JK(x-y)h(y)dy
from which (2.61a) can be derived.]
(c) Solution of a mixed problem by Fourier transformation
In many cases mixed initial-boundary-value problems can be solved by
Fourier transformation, when the domain of the solution is a half space.
The method will be illustrated (following R. Hersh) by a problem for the
wave equation for n = 3, which could also be solved by reflection. (Compare problem 3, p. 119.) We seek to find a u(x,t)=U(X 1,X2,x3,t) for which
Du=O for X3>0, t>O
u= ut =0 for X3 >0, t=O
Mu= ut + a1ux , + a 2uX2 + a3uX3 = h(X 1,X2,t) for X3=0, t >0.
(2. 62a)
(2.62b)
(2.62c)
We assume here that the ak are constant, that a3 <0, and (to avoid
inconsIstencies for t = X3 = 0) that there exists a positive e such that
h(X 1,X2,t)=0 for t<e.
(2.62d)
133
5 Hyperbolic equations in higher dimensions
Let moreover h E CO (jR3) with a sufficiently large s.
The building blocks are again exponential solutions
u = ei(lu+€,x, +€2 X2+ €3 X3)
(2.63a)
of (2.62a), for which the relation
(2.63b)
A2-c2(M+M+~n=0
will have to hold. For these u we have
Mu= i{A+ al~1 + a2~2 + a3~3)ei(lu+€,x,H2X2)
for x3 =0.
(2.63c)
This leads to a formal solution of (2.62a,c) given by
_
-3 / 2 f ei(lu+€,x, +~2X2+€3X3)
~
u{x,t)-{2w)
.(A
~
~
~ ) h (~I'~2,A)d~ld~2dA.
1
+a 1 1+a2 2+ a 3 3
Here
Ii is the Fourier transform of h:
Ii (~1'~2,A)=(2w)-3/2 e-i(lu+~,x'+€2x2)h{Xl,x2,t)dxldx2dt
f
(2.64)
(2.65)
and ~3 in (2.64) is a function of (~1'~2,A) satisfying (2.63b).
For convergence of the integral in (2.64) it is essential that the exponential in the integrand for each fixed t is bounded for x in the half space
X3>0. This is the case, when ~1'~2 are real and
Im~3
>o.
(2.66)
This condition does not fix the solution ~3 of (2.63b) uniquely for all real
We also have to worry about possible vanishing of the denominator of the integrand. It is best to shift the integration with respect to A in
(2.64) in the complex plane, letting A run through values with ImA = - 8
with a fixed real number 8 > o. Under these circumstances the solution ~3
of (2.63b) cannot be real, and there is a unique solution ~3' for which (2.66)
holds. Moreover the denominator in (2.64) cannot vanish since
~1'~2,A.
Im(A + al~l + a2~2 + a3~3) = - 8 + a 3Im~3 < - 8,
(2.67)
using (2.66) and the important assumption a3 < O. We obtain the estimate
(2.68)
for the absolute value of the integrand in the expression (2.64) for u. We
now make use of assumption (2.62d), which implies that
in the integral (2.65) giving Ii. We conclude, similarly as in (2.47), that for
the complex A and real ~l' ~2 in question
(2.69)
with a suitable constant M. Thus for sufficiently regular h of compact
134
2 Higher-order Hyperbolic Equations with Constant Coefficients
support formula (2.64) furnishes an actual solution of (2.62a,c). It remains
to show that it satisfies (2.62b) as well. Now by (2.68), (2.69)
MT f (1 +a+M+ IAI2fs d~ld~2dA.
lu(x,t)1 <(2'1l-)-3/2
Letting
8-HX)
8(/-e)
it follows that
u(x,t)=O for X3>0, 0< t<e.
This implies that (2.62b) holds.
(d) The method of plane waves
In what preceded the standard initial-value problem was solved by decomposing the initial function g(x) into exponentials exp(ix·~, according to
Fourier's formula. For those the initial-value problem is easily solved; by
superposition we then obtained the solution for general g. A disadvantage
of this method is that the resulting solution u is expressed in terms of the
Fourier transform g instead of directly in terms of g. For homogeneous
partial differential equation with constant coefficients a different type of
decomposition of g into plane waves can be preferable, since it does not
involve the somewhat artificial introduction of exponentials.
A function G with domain R n is called a plane wave function, if its level
surfaces form a family of parallel planes, that is if G can be expressed in
the form G= G(s), where the scalar argument
s=x·~= ~~kXk
(2.70)
k
is a linear combination of the independent variables. The exponential
functions above are plane wave functions with G=e is • Assume that the
differential operator P(D,T) is homogeneous of degree m, and thus agrees
with its principal part Pm(D,T). Let P(O, 1)= 1 and the degree m of P be
even. We notice that for any function G(s) of a scalar argument s we have
in
u(x,t)= G(x·~+At)
(2.71)
P (D,T)U = P (~,A)G(m)(x·~ +At).
(2.72)
a solution of
In particular u will be a solution of
P(D,T)U=O
if
~,A
(2.73a)
satisfy the algebraic equation
P(~,A)=O.
(2.73b)
We can find a linear combination of expressions (2.71) corresponding to
the various roots A=Ak of (2.73b), satisfying the standard initial conditions
135
5 Hyperbolic equations in higher dimensions
for t=O
rku=O
rm-1u= g
for k=O, 1, ... m-2
(2.74a)
for a plane wave function
g=g{s)=g{x'~)'
(2.74b)
In the case where g(s) is an entire analytic function of s we easily verify
that a solution is given by the Cauchy integral
I
u{x,t)= 2'1Ti
J(G{x·~+i\t)
P{~,i\) di\,
r
(2.75a)
where the path of integration r in the complex I.-plane encloses all roots i\k
of (2.73b). Here G is to be chosen so that
G<m-I){s)=g{s).
(2.75b)
In the special case G (s) = eis we regain the formula
u{x, t) = eix·I;Z (~, t)
(2.75c)
with Z given by (2.31).
We now restrict ourselves to the case that P is strictly hyperbolic, that is,
that all roots i\k of the homogeneous equation are real and distinct for
~ =1= O. In that case the calculus of residues permits us to evaluate the
integral (2.75a) for ~=I=O:
_ ~ G{x·~+i\kt)
u{x,t)- .L.J
.
k=! P~{~,i\k)
(2.76)
Now formula (2.76) was derived under the assumption that G is analytic.
By continuity arguments or direct verification it follows immediately that
(2.76) represents a solution of (2.73a), (2.74a,b), when G is only of class
cm(~) and satisfies (2.75b). Of special interest for us will be the case when
g is of the form
(2.77a)
depending on parameter vectors
(2.73a), (2.74a,b) is then given by
_
U{x
y,~.
The corresponding solution u of
_ ~ ({x-y)·~+i\kt)msgn({x_y)·~+i\kt)
'P (t I.)
k=!
m. ~ <;;, k
y,t,~)-.L.J
(2.77b)
where we have used (2.76)*
G{s)=
·Strictly speaking here
harmless.
136
G~cm(R).
(s- y.~)msgn{s- y.~)
,
.
m.
But the jump discontinuity in G(m)(s) for
(2.77c)
s=y'~
is
2 Higher-order Hyperbolic Equations with Constant Coefficients
We shall show that for odd n and for s sufficiently large the general
g(x)E Cooo(Rn) can be decomposed into functions of the form (2.77a) with
I~I = 1 by a formula of the type
g(x)=
f..IEI=ldS(J 4Y1(x-y)-~lq(y)
(2.78a)
with a suitable continuous function q(y) of compact support. It follows
then that
u(x,t)=
f..IEI=ldS€J 4Y U(x-y,t,~)q(y)
(2.78b)
solves our standard initial-value problem (2.14a,b,c).*
To arrive at (2.78a) we first decompose the function
r=lx-yl
into plane waves of type (2.77a). This is achieved by writing
x- y
= rq,
where 1111 = 1.
(2.79a)
(2.79b)
Then
(2.79c)
Here Cn is a positive constant independent of 11, since a simultaneous
rotation of 1I,~ does not change the integral of lI'~ over the unit sphere, so
that we can always bring about that 11 is the unit vector in the xn-direction.
By (1.9) of Chapter 4,
i::.xri = j(j + n - 2)ri-2.
It follows for odd n that there is a constant dn such that
d i::.(n-l)/2r =
n x
2-n
r
= K(x,y)
(2- n)wn
(2.80)
is a fundamental solution of the Laplace equation with pole y (see Chapter
4, (1.l5a». Equivalently
!.
k(x,y)=dnr=cn-1dn
I~I= 1
l(x-Y)'~ldS
(2.81)
is a fundamental solutiont with pole x for the operator ~n+l)/2:
g(x)=
f k(x,y)i::.~n+l)/2g(y)4Y.
(2.82)
Substituting for k its expression (2.81) we are led to a decomposition
*Tbe U given by (2.77b) has continuous derivatives of order .;;; m with respect to its
arguments, except for jumps in the mth derivatives. We use here that the P).. in the
denominators are bounded away from 0 for I~I = 1, because P(~,A) has no multiple roots.
tPormula (2.82) follows rigorously from (1.31) of Chapter 4 by integration by parts, provided
gE q (IRn) with s;;. n+ 1.
137
5 Hyperbolic equations in higher dimensions
(2.78a) for g with
(2.83)
With this q, formula (2.78b) solves the standard initial-value problem.
PROBLEMS
1. Give the value for n = 3 of the constants
[Answer: c3=2'17, d3= -1/8'17.]
Cn>
d"
~ formulas
(2. 79c), (2.80).
2. Show that (2.78b) can be rewritten in the form
u(X,t)=Cn-Id,,!~g(y)dy
r
J1cl =!
~U(x-y,t,~)dSc'
(2.84)
where
m)
. (n+l
k=mm
-2-'2 '
j=n+l_ k
2
(2.85)
3. For the wave equation (1.1) in n=3 dimensions identify the solution obtained
from (2.84) with that given by formula (1.14). [Hint: Show that here
i(x-yHi<ct
for i(x-yHi>ct
for
flyU=l/c
flyU=O
i1€1=1
-
-
flyU(x y,t,~)dS~- {4'17t1i
4 1 x- yi
'17 c
for ix-yi >ct
for ix-yi <ct.
(2.86)
Apply Green's formula.]
4. Show that for n = 3 dimensions formula (2.82) for g E CJ(JR3) takes the form
g(x) = -
~flx
16'17
r
JI~I=!
dSc!i(x-yHiflyg(y)dy.
(2.87)
Show that here
where G(~,p) (the Radon transform of g) denotes the integral of the function g(y)
over the plane Y'~=p in JR3. Formula (2.87) solves the Radon problem of
determining a function from its integrals over planes (for the case when g has
compact support). Show how to obtain a different solution formula for this
problem from Fourier's formula (2.15), (2.16) by expressing the Fourier transform g in terms of the Radon transform G.
138
3. Symmetric Hyperbolic Systems
(a) The basic energy inequality
In this section we shall be concerned with a linear first order system of
P.D.E.s for a column vector u=u(x,t)=u(xl, ... ,xn,t) with N components
U 1•• •• , UN' Such a system can be written symbolically in the form
n
Lu=A{x,t}ru+ ~ Ak{x,t)Dku+B{x,t)u=w{x,t).
(3.1a)
k=1
Here A,A I, ... ,A n,B are given N x N square matrices, w a given N-vector,
and 7',D 1, ... ,Dn again stand for the differential operators
7'=
a
at'
a
DI = aX I
'
... ,
a
Dn= axn '
As initial data we prescribe the values of u on the hyperplane t = 0 in
xt-space. By a trivial substitution on u and w we can always bring about
that the initial condition becomes
(3.1b)
u{x,O)=o.
Following K. O. Friedrichs the system (3.1a) is called symmetric hyperbolic, if all of the matrices A,A1, ... ,A n (but not necessarily B) are
symmetric, and moreover A is positive definite for all arguments (x, t) in
question. We shall see that symmetric hyperbolic systems (for sufficiently
regular A,A\ ... ,An,B,w) are indeed hyperbolic in the sense that the
initial-value problem (3.la,b) can be solved.
Many hyperbolic equations or systems can be reduced to symmetric
hyperbolic form. Consider, for example, a single scalar hyperbolic secondorder equation
n
n
Vtt = ~ aik {x, t)VX;Xk + ~ bi{x,t)vx,+c{x,t)v,+d{x,t)v,
~k=1
i=1
(3.2)
where the aik form a positive definite symmetric matrix. We introduce here
the vector u with the N = n + 2 components
(3.3)
... ,
The n + 2 equations
n
n
~ aik7'Uk- ~ aik Dk Un + 1 =0
k=1
k=1
n
7'Un+l -
fori=l, ... ,n,
n
~ aikDkui - ~ biui-cun+l-dun+2=0,
i,k=1
(3.4a)
(3.4b)
i=1
(3.4c)
139
5 Hyperbolic equations in higher dimensions
are consequences of (3.2), (3.3). One easily verifies that they constitute a
symmetric hyperbolic system. Similarly in Chapter 2 we were able for n = I
to write general hyperbolic systems (3.la) in "canonical form," where A
becomes the unit matrix and A J a diagonal matrix. This clearly implies a
reduction of the system to symmetric hyperbolic form.
Multiplying (3.la) with the transposed vector uT we find that*
n
T(uTAu)+ ~ Dk(uTA ku)+u TCu=2u Tw,
(3.5a)
k=l
where
n
C=2B-TA - ~ DkA k.
(3.5b)
k=l
Integrating (3.5a) over a region R in xt-space and applying the divergence
theorem yields
[:T(
A:
+ ~>
<:: )udS~ 1.{-
uTeu + 2uTw)dx dt,
(3.6)
where dx J / dp, ... , dXn/ dp, dt / dp denote direction cosines of the exterior
normal and dS the element of "area" of the boundary aR, while dx dt =
dxJ ... dxndt is the element of volume of R.
We define the slab R"h in xt-space for r..>O as the set
R"h={(x,t)lxE~n,O<t<r..}.
(3.7)
Let for a certain T the function u(x, t) be a solution of (3.la,b) of class
CJ(R T ), which is of compact support in x for each t in 0 < t < T. Applying
(3.6) to R = R"h with 0 < r.. < T yields the energy identity
E(r..)=f
t=A
uTAudx= fAdtf( -u TCu+2u Tw)d.x.
)0
(3.8)
We assume that the matrices A1A i, ... ,A n,B together with their derivatives
of any desired order are continuous and bounded uniformly in R T •
Moreover the matrix A shall be uniformly positive definite in the sense that
there exists a /L > 0 such that
(3.9)
for all (x,t) in RT and all vectors v. Since C is bounded there will exist a
constant K > 0 such that
(3. lOa)
·We make use here of the symmetry of A,A I, .•. ,A n which implies
T(uTAu) =( TU1)Au + uT( TA)u + uTA (TU) =2uTA (TU) + uTe TA)u
and analogous identities for Dk(uTA "it). We tacitly assume thatA,A I, .•. ,A n are in C l , and B
and w in Co.
140
3 Symmetric Hyperbolic Systems
for (x,t)ERT and all v. Moreover, since A is symmetric and positive
definite the inequality
2uTw < I-"UTu+ ~WTW < uTAu+ _1- wTAw
I-"
1-"2
(3. lOb)
holds for all vectors u, w. Thus by (3.8)
E(A)«K+l)L h E(t)dt+I-"-2J. wTAwdxdt forO<A<T.
o
RA
Writing this as
(3.11)
.!£e-(K+I)"" r""E(t)dt< e-(K+I)"" 1-"-2 r wTAwdxdt
dA
Jo
JRr
we conclude that
io
T
e(K+I)T -1
J.
E(A)dA< K 1 1-"-2 wTAwdxdt
+
Rr
< JL- 2Te(K+I)T J. wTAwdxdt.
(3.12a)
Rr
We define the inner product of two vectors u, v on RT by
(u,v)= J. uTAvdxdt,
and denote by II u II =
y( u, u)
Rr
(3.12b)
the corresponding norm. Setting
(3.l2c)
the estimate (3.12a) takes the form
(u,u) < r2(w, w),
(3.13)
whenever u is a solution of (3.la,b) of compact support in x.
Denote by ~s the space of functions uE CS(R T ), that vanish on t=O,
and have compact support in x. Then by (3.13) the energy inequality
(3.14)
holds for all UE~I.
The estimate (3.14) can be made the basis for an existence proof for the
solution of the initial-value problem (3.1a,b). A first step in this direction is
to establish the existence of a weak solution of that problem. For that
purpose one introduces in analogy to ~ 1 the space (;1 of functions
vEe 1(RT ) that are of compact support in x and vanish for t = T. For any
u E ~ I, v E (;1 we derive by integration by parts Green's identity (see (4.5)
of Chapter 3)
(v,Lu) =(iv,u)
(3.15)
141
5 Hyperbolic equations in higher dimensions
where
i, the adjoint of L, is defined by
n
iv= -T(Av)- ~ Dk(Akv)+BTv
k=l
= -Lv+(CT +B-BT)v
(see (3.5a,b». A function u E C 1(RT) is a solution of (3.1a,b) if and only if
(v,w)=(iv,u)
(3.16)
for all v E CI. For by Green's identity (not using (3.1 b), but using that v
has compact support in x and vanishes for t = T)
(iv,u)=(v,Lu) + J=ovTAudx.
(3.17)
Taking for v in (3.16) first an arbitrary function of compact support in RT
and vanishing for t = 0, we see that u satisfies (3.1a). Subsequently we find
from (3.16) that
o=f vTAudx
t=O
for all v E Cl , which implies (3.1 b).
This suggests replacing (3.1a,b) by the requirement (3.16) in some
suitable function space. For that purpose we observe that C l is an inner
product space (see p. 95), if we define as inner product of two vectors v
and v' the expression
(v, v') = (iv,iv').
(3.18a)
Indeed this expression is linear and symmetric in v,v'. One only has to
verify that the square of the corresponding norm 111·111 satisfies
IIlvIW=(v,v)=(iv,iv»o
(3.18b)
for v*O. This is obvious, since - i is again a symmetric hyperbolic
operator; replacing t by T - t the class Cl goes over into {; I. It follows in
analogy to (3.14) that there exists a constant f such that
-2
-
-
-2
(v,v)<r (Lv,Lv)=r (v,v),
(3.19)
which implies (3.18b). We complete Cl into a Hilbert space H, by taking
Cauchy sequences of functions in Cl with respect to the norm Illvlll.
Because of the inequality (3.19) we have for v in Cl and wE CO(RT ) that
I(v, w)1 < I/vllllwi/ < fi/wl/lllvlll·
Since by (3.19) Cauchy sequences with respect to the norm Illvlll also are
Cauchy sequences with respect to the norm II vii, it follows that (v, w)
defines a bounded linear functional on H. By the representation theorem
(p. 96) we can then find an element U in H such that
(v, w)=(v, U)=(iv,iU).
142
3 Symmetric Hyperbolic Systems
Obviously then u=iu satisfies (3.16) for all vEH. This u, which belongs
to the set of square integrable functions in R T , can be considered a weak
solution of the initial-value problem (3.la,b). It remains, of course, to show
that u can be identified with a strict solution in the ordinary sense, at least
for sufficiently regular w. Here we shall not go into a proof of this fact,
given in a classical paper by K. o. Friedrichs, * but shall instead give below
an existence proof based on a completely different approach (also due to
Friedrichs), namely the method of finite differences.
In connection with proving regularity of solutions it is important to
have "a priori" estimates for the values of a solution u and of its
derivatives at each point (x, t) of RT" Such estimates can be obtained by
first deriving estimates for the integrals of the squares of a sufficient
number of derivatives. Those in tum can easily be obtained by using the
differential equations satisfied by the derivatives. We find from (3.1a) and
any i= I, ... ,n that
Djw=DjLu= LDju + (DjA}ru + ~ (DjAk)Dku+(DjB)u.
(3.20a)
k
Substituting still for 'TU its expression from (3.la) we see that Dju satisfies
an equation of the form
LDju= ~aikDku+biu+Diw+ejW
(3.20b)
k
with certain square matrices aik,bi,ej, which by assumption are bounded in
RT" There exists then an M such that
IILDjul1 <
M{; IIDkuli + lIull + IIWII) + IIDjwll·
(3.20c)
Assume that UE~2. Then DiUE~I, and we find from (3.14) that
IIDiul1 <rM(; IIDkUII+llwll)+Mr21Iwll+flIDiWII.
Summing over i yields
~ IIDiUl1 <rMn~ IIDkull+nMr(I+r)llwll+ ~ IIDiWII.
k
j
(3.20d)
i
Now by (3.12c) the constant r can be made arbitrarily small by choosing T
sufficiently small. Hence for sufficiently small T we have r Mn < 1/2, and
conclude from (3.20d) that
~ IIDiUl1 <2nMr(I+r)lIwll+2nr~ IIDjWII·
i
i
(3.21)
A similar estimate can then be obtained for II'TUIl directly from (3.la).
* Symmetric hyperbolic linear differential equations, Comm. Pure Appl. Math. 7, (1954),
551-590.
143
5 Hyperbolic equations in higher dimensions
Forming next second space derivatives of (3.1a) and proceeding in this
manner, we find that for sufficiently small T we can estimate all DClzl for
11001I<s in terms of the IID.Bwll for lfil<s (using, of course, the existence
and uniform boundedness of the derivatives of the coefficients of L).
Moreover by (3.11), (3.12b) we have for any UE~I and any A between 0
and T that
[=>.uTAUdx=E(A) «K + 1)lluIl 2+ p. -21ILuIl2.
Then for u E ~s by (3.9)
[=>.(DClzl)T(DClzl)dx < p. [=>.(DClzl)TA (DClzl)dx
-I
< p. -I(K + 1)IIDClzlI1 2+ p. -311 LD Clzl11 2.
This inequality permits us to estimate for any component uj of u and any 0:
with 10:1 < s
[=>.IDau/dx
in terms of* the IIDflwll with 1131 < s.
The transition from estimates for integrals of squares to pointwise
estimates is furnished by one of the Sobolev inequalities. This inequality
can easily be derived from Parseval's identity, known from the theory of
Fourier integrals, that connects the square integrals of a function g with
that of its Fourier transform g:
(3.22)
Let
s=[l]+I,
(3.23)
where generally [v] for a real number v denotes the largest integer < v.
Thus s is the smallest integer exceeding n12. By (2.16) and the
Cauchy-Schwarz inequality
g2(x)1 = (2.n
1
< Cn
rn \f eix'~{l + I~I)-s{l + 1~lyg (~)d~\
2
f (I + I~D2sl g(~)12d~,
where
cn=(2'IT)-n
f {l + IW-
2s
d~< 00.
·We use here that LD"u differs from D"w by a differential operator of order JaJ.
144
3 Symmetric Hyperbolic Systems
By (2.20)
(1 + 1~1)2'1 g (~)12..; 22sn' ( }:
lal';;;'
l(i~tg (~)1)2
~,
";dn }: 1(i~tg(~W=dn }: ID ag1 2 ,
lal.;;;'
lal';;;'
with
dn = 22sn' }: 1.
lal.;;;s
Using (3.22) with g replaced by Dag it follows that
Ig2(x)l..; cndn L
lal';;;s
jIDag (Y)1 2dy.
(3.24)
Applied to the function g= u(x,t) for a fixed t between 0 and T, inequality
(3.24) permits us to estimate lu(x,t)1 in terms of the IID.Bwll with 1.81..; s,
with s given by (3.23).
PROBLEMS
1. (a) Write the system (3.4a,b,c) in matrix notation and show that it is symmetric
hyperbolic.
(b) Do the same for Maxwell's equations (Chapter 1, (2.6a).
2. Show that a symmetric hyperbolic system (3.1a) with constant A,A 1, ... ,A nand
B =0 satisfies the Girding hyperbolicity condition.
3. (Local uniqueness for the initial-value problem (3.1a,b).) Let
bounded set in IRn, described by a function </>(x) for which
</>(X»O for xEw,
w
be an open
</>(x)=O on ilw.
(3.25a)
for A> 0 let R>. denote the "lens-shaped" region in xt-space given by
(3.25b)
xEw,
and S>. denote the hypersurface
xEw,
t=A</>(X),
(3.25c)
so that R>. is bounded by S>. and SQ, Set
(3.25d)
(a) Show that for a solution of (3.1a,b)
E(A)=
r
JR)..
(-u T Cu+2uT w)dxdt.
(3.25e)
145
5 Hyperbolic equations in higher dimensions
(b) Call S" spacelike if the quadratic form in v
Q" (v)= VT(A (x,/) - p.
'± A k(X,/)q,Xk (X»)V
(3.25f)
k=l
is positive definite for all (x, I) E S,.. (This is certainly the case for sufficiently
small p..) Assume that the S" are spacelike for 0 < p. < A. There exists then a
K such that
(3.25g)
for all v and all (x,/)ERA' taking p.=I/q,(x). Show that for a solution of
(3.la,b) with w=O
(3.25h)
(c) Show that a solution of (3.la,b) is uniquely determined in RA by the values
of w in RA, if the S" are spacelike for 0 < p. < A. Compare this result with
Holmgren's theorem, p. 66.
4. Write the wave equation Ou=O as a symmetric hyperbolic system. Show that
the definition of a spacelike surface using the form (3.25f) agrees with the one
given by (1.54).
5. (a) Prove Sobolev's inequality (3.24) for n = 1 with cndn replaced by 2 from the
identity
(3.26a)
(b) Prove by induction over the dimension n that for X=(XI'''''Xn )
g2(x)<2n
L f(D g(y»2dy.
a
(3.26b)
lal«n
(b) Existence of solutions by the method of finite differences*
We follow here the ideas and notations used on a trivial example in
Chapter 1, p. 4. We cover (n+ I)-dimensional xt-space by a lattice. We
take three positive quantities h, k, T, fixed for the moment, and consider
the set ~ of points (x, t) for which
x = (x], ... ,xn) = (a]h, ... ,anh),
t = mk, 0 < t < T.
(3.27)
Here a], ... , a" , m shall be integers. It is convenient to combine the aj into a
multi-index Ii = (a l , ... , an), where "-,, shall indicate that the components aj
of Ii range ovt'lr all integers, in contrast to the common multi-indices a
whose components shall continue to be nonnegative. Then ~ consists of
the points
x=lih,
* ([18], [25D
146
t=mk withO<m< T/k.
(3.28)
3 Symmetric Hyperbolic Systems
We define operators
Eo,~
corresponding to shifts to neighboring points:
~U(XI,.",Xn,t)= U(XI' ... ,xj + h, ... ,xn,t)
forj= 1, ... ,n,
Eou(x l , ••• ,xn,t) = u(x l , ••• ,xn,t+ k).
(3.29a)
(3.29b)
Obviously ~ has the inverse ~ - I, where
(3.29c)
More generally combining (E1, ... ,En) into a symbolic vector E, we can
write
E"u(x, t) = u(x + iih, t) = U(XI + a1h, ... 'Xn + a"h,t).
(3.29d)
We next define divided difference operators 80, ~ by
~=
E.-l
T
forj= 1, ... ,n
Eo-1
80 = - k -
(3.30a)
so that, for example,
80 u(x,t) =
U(X,t+ k) - u(x,t)
k
.
(3.30b)
For functions with continuous second derivatives these difference
quotients approximate the corresponding derivatives, and we have by
Taylor's formula
~U(X,t) = Dju(x,t) +O(h),
80 u(x,t) = Tu(x,t)+O(k).
(3.31)
It would appear natural to replace the differential equation (3.la) for the
vector U by the difference equation
n
A80 v+ ~ A j8j v+Bv=w.
(3.32)
j=1
However in order to ensure stability, we have to select a more complicated
difference scheme. We replace the space derivatives Dju by central difference quotients (2h)-I(~ - ~ -I)V, and, following Friedrichs, use instead
of (3.32) the system of difference equations
l(
In
)
In.
AV="kA Eo- 2n ~ (~+~-I) V+ 2h ~ AJ(~_~-I)V+Bv=w.
J=I
J=1
(3.33a)
Here it is understood that the argument of all functions is the same point
(x, t), only shifted as indicated explicitly. The operator A is defined by
147
5 Hyperbolic equations in higher dimensions
°
(3.33a). Equation (3.33a) is to hold for those (x,t) for which (x,t) and
(x, t + k) belong to ~, restricting t to the interval ~ t ~ T - k. Since the
matrix A(x,t) is nondegenerate, we can solve (3.33a) for Eov=v(x,t+k)
in terms of w(x,t) and the v(y,t) with
~ IYi- Xii ~ h.
i
Thus formula (3.33a) constitutes a recursion formula that permits us to
find v at the time t+ k if v and ware known at the time t. If we add the
initial condition
v(x,O)=o,
(3.33b)
then there exists trivally for given w a unique solution v(x,t) of (3.33a,b) in
Obviously the value of v at any point (x, t) is determined already by the
values of w(y,s) at a finite number of points (y,s), namely those for which
~.
(3.34)
The domain of dependence of vex, t) on w here has the shape of a pyramid
with vertex (x,t- k). If, as we shall assume, w(y,s) has compact support in
y for each s, then vex, t) has compact support in x.
We are relieved of the burden of proving existence for v. Instead we
shall have to show that v for h,k~O can be used to approximate a function
u(x,t) defined in R T , which is a solution of (3.la,b). For that we shall need
the "discrete" analogues of the energy inequality (3.14), of similar inequalities for higher derivatives, and of Sobolev's inequality. We shall make use
of the same symmetry and regularity assumptions on the matrices
A,A', ... ,An,B as before on page 140.
We write (3.33a) as
n
AEov= ~ (aiEi+biE.i-I)v-kBv+kw,
(3.35a)
j=l
where the matrices ai,IY' are defined by
.
1
k·
aJ=-A--AJ
2n
2h'
(3.35b)
The matrices aj, hi are symmetric. Since A is positive definite, the same
holds for ai, hi provided the ratio k / h is sufficiently small. More precisely,
using the fact that the Ai are bounded and A is uniformly positive definite
there exists a positive A (held fixed in what follows) so small that the
matrices ai, hi are positive definite for all (x, t) in R T , when the space and
time steps are connected by
k=Ah.
(3.36)
Now for a positive definite symmetric matrix a and any vectors v, w the
148
3 Symmetric Hyperbolic Systems
inequality
2vTaw<2YvTav YWTaw <vTav+wTaw
(3.37)
holds.* (Compare (3. lOb).) Multiplying (3.35a) by 2EovT from the left, and
using (3.37) for a = aj, b j we get
2(EovlA(Eov)
«EoV)TC~1 (a j +~) ) (Eov)
n
+ ~ ((~v)Taj(~v)+(~-lv)T~(~-Iv))
j=1
(3.38)
By (3.35b)
n
L (aj+~)=A.
(3.39)
j=1
Moreover
(~v)Taj (~v)= ~(vTajv) - (~v)T(~aj - aj)(~v)
= ~(vTajv) - h(~V)T( ~aj)(~v).
Now by assumption the derivatives, and hence also the difference
quotients, of the aj are bounded uniformly for (x,t)ERT" Using (3.9), we
can (after applying E) find a constant K such that
(~v)T(8jaj)(~v) < K(~v?(~A)(~v)=K~(VTAv).
Thus t
Similarly
( ~ - IV) T ~ ( ~ -
IV)
= ~ - I ( v Tbjv) + O( h~ - I ( VTAv) )
(Eovl A (Eov) = Eo(vTAv) +O(kEo(vTAv»
2k(Eov)T(Bv - w)=O(kEo(vTAv) + k(vTAv) + k(wTAw»).
* For a=urut matrix, relation (3.37) is just the Cacuchy-Schwartz inequality + the elementary
inequality 2xy " x 2 + y2 valid for real numbers x,y. The case of more general a goes over into
the previous one by writing a=cTc with a suitable matrix c. This amounts to writing the
positive definite quadratic form vTav as a sum of squares.
tWe use the customary notation F=O(G) to indicate that there exists a constant K such that
1FI .;; KG for all quantities F, G in question.
149
5 Hyperbolic equations in higher dimensions
We then find from (3.38) that
n
Eo(vTAv)= ~ (~(vTajv)+~-I(VTbiv»
j-I
+0
[{~l (Ej + Ej-I)(vTAv)
+ k(vTAv) + kEo (vTA v) + k(WTAW))]. (3.40)
We sum this inequality for a fixed 1 = mk over all x of the form ah. In
analogy to the E (A) of (3.8) we introduce the energy sum
.,,(1) =hn~ vT(iih,I)A (iih,I)V(iih,I) = h n~ E" (vTAv),
(3.41)
where ii ranges over all vectors with integers as components. Define
similarly
(3.42)
Since, by assumption, w(x, I) is of compact support in x, our sums only
contain a finite number of nonvanishing terms. Obviously we arrive at the
same sum in (3.41), if vTAv is replaced by ~(vTAv) or ~-l(VTAv), since
such a shift does not affect* the set of points over which we sum vTAv.
Thus we obtain from (3.41), (3.40) that
n
.,,(/+k)=h n ~ (vTajv+vTbiv)
j=1
+ 0 (h.,,( I) + k'f/(I) + k'f/(I+ k) + kr (I»
< .,,(/) + K(h +-k).,,(I) + k'f/(I + k) + kr (I»
(3.43)
. with a certain constant K. If k=Ah is so small that Kk< t, we can solve
(3.43) for .,,(t+ k), and obtain an inequality of the form
.,,(I+k) < eC~(I)+kyr(l)
with certain constants C,y. Since .,,(0)=0, it follows for I=mk < T that
.,,(/) < kyU (t- k)+ eCkr (t-2k)+··· + e(m-l)kcr(O»)
m
< kye CT ~
,,-0
r(pk).
(3.44)
*This observation plays the same role as did the integration by parts in the derivation of
(3.8)_
150
3 Symmetric Hyperbolic Systems
In analogy to (3.12b) we define the norm IIwll of the vector w(x,t) for the
lattice ~ by
IIwll 2=h nk ~
(w(X,t»)T A(x,t)w(x,t)=k
(x,t)E~
~
O<m< T/k
r(mk). (3.45)
Summing (3.44) over all m with 0 <;; m <;; T / k we obtain the energy estimate
IIvll2 <;; yTeCTllwll2= yTe CT IIAvl1 2
(3.46)
in analogy to (3.14), (3.12c).
The next step consists of deriving similar energy estimates for the
difference quotients of v. (Compare (3.20a, b, c, d).) These are obtained by
applying the operator 8, for r= 1, ... ,n to (3.33a). We make use of the rule
for differencing a product of two functions U, V
8,(UV)=(E,U)(8,V)+(8,U)V
(3.47a)
and of the identity
i
(Eo- 21n
j~1 (~+ ~-I) )v= (80-
21/\ -I
j~18Al- ~-I) )v,
21h (~- ~ -1)= !(1 + ~-1)8j'
(3.47b)
(3.47c)
We then find for 8, v an equation of the form A8, v = w' where w' is a
linear combination of 8, w, wand the 8s v, ~ - 18s v with s = 1, ... , n. Applying
the estimate (3.46) to 8,v instead of v, and summing over r, we arrive at an
inequality of the type
(3.48)
provided T is sufficiently small. Repeating this procedure we arrive at
estimates for the norms of the higher difference quotients:
~
lal<s
118avll2 = 0 (
~ 118~112),
(3.49)
lal<s
where we have combined the operators 81, ... ,8n into a vector 8, and write
(3.50)
Using (3.49), (3.9), we find then for any t = mk between 0 and T and any a
with lal <;; s that also
h n ~ (8 av( t3h,t))T( 8av( t3 h ,/)) <;; !l-Ihn~ (8 aV)TA (8 av)
= o (IIA8 avI1 2) = 0 (
~
11'1<s
IJ8YwI12).
(3.51)
151
5 Hyperbolic equations in higher dimensions
From the i 2-estimates (3.51) we pass to pointwise estimates by developing a difference analogue to Sobolev's inequality (3.24). This we can derive
without recourse to Fourier transforms. (Compare problem 5, p. 146.) We
start with the case n = 1 of a function g(x) of a scalar argument x. For a
nonnegative integer r we have the identity
r-l
g(x)=g(x+rh)- ~ (g(x+(P+ I)h)-g(x+ph))
1'=0
r-l
=g(x+rh)-h ~ 8g(x+ph),
1'=0
where, of course, 8g(x) stands for (g(x+h)- g(x»/h. Squaring we get the
estimate
r-l
g2(x),2g2(x+rh)+2rh2 ~ (8g(x+ph))2
1'=0
by Cauchy-Schwartz. Summing over r=O, I, ... ,p-I yields
00
00
pg2(x),2 ~
r= -00
g2(x+rh)+p 2h2 ~ (8g(x + rh))2.
r=-oo
We choose here for p the integer determined by
1
I
h'P<h+1.
For h sufficiently small, say h < V2 - I, we have p 2h2< 2, and hence
00
~(x),2h ~ (g(x+rh)f+(8g(x+rh)?
r= -00
(3.52)
For x that are multiples of h we can replace x by 0 on the right. Next for a
function g(x 1,X2) we find by repeated application of (3.52) that
00
g2(X),X2),2h ~
[(g(r)h,x2))2+(8)g(rlh,x2))2]
rl= -00
,4h 2
00
~
[(g(r)h,r2h)f+(8)gi+(82g)2+(8)82g)2].
'1"2=-00
Generally for any n and for x of the form yh
g2(X) , 2nh n ~ ~ (8 ag(,Bh)t
lal<n ii
It follows from (3.51) that for a solution of (3.33a, b) in
18 av(x,tW= 0 (
~
Ipl<lal+n
provided T (for given a) is sufficiently small.
152
118PwI12),
(3.53)
~
and any a
(3.54)
3 Symmetric Hyperbolic Systems
We shall need estimates of the type (3.54) which do not depend on the
particular values h which we shall let tend to O. For that purpose the
expression IIB.Bw1l defined by (3.45) as a sum over ~ involving difference
quotients of W will have to be replaced by an integral over RT involving
derivatives of w. Let w have components WI' ••• ' Wn- Since A is bounded we
have
IIB.BwII=hnk ~
(8.Bw(X,t»TA (X,t)(B.Bw(x,t»
(x,t)E~
~
=O(hnk"i:
r=1
(8.Bwr(X,t»2).
(3.55)
(X,t)E~
For the scalar Wr we have by the mean value theorem
min DjwrCy,t)";; ~wr(x,t)";; max Djwr(y,t).
Iy-xl';;h
It follows by induction for
min
Iy-xl.;; sh
Iy-xl.;;h
IfJl..;; s that
D.Bwr(y,t)..;; 8.BwrCx,t)";;
max D.Bwr(y,t),
Iy-xl.;;sh
that is, that
Bfiwr(x,t) = DfiwrCy, t)
for some y with
Iy - xl..;; sh. Thus
hnk ~
(8 fiwr(x,t»2
(x,t)E~
is essentially a Riemann sum for the integral
1 (D.Bwr(x,t»2 dxdt.
RT
More precisely, for wr of classes and of compact support in RT the sum
will converge to the integral for h---70. Consequently by (3.54)
max I8"V(X,tW =O(
(x,t)E~
L
IPI.;;lal+n
1,D.Bw(x,tWdxdt)
(3.56)
RT
for all sufficiently small h. We get similar estimates for the mixed spacetime difference quotients B~Ba by solving (3.33a) for Bov, using (3.4Th, c).
In what follows we shall assume that w(x,t) is of class C n + 2(RT) and of
compact support in x. Then v and all its difference quotients of orders ..;; 2
are bounded on ~ uniformly independently of h. This implies that v and its
difference quotients of orders ..;; I are uniformly Lipschitz on ~, with a
Lipschitz constant that does not depend on h.
The rest is simple. We now refine our lattice ~ indefinitely, choosing
h=2- q, k=A2-q with q=I,2,3,4, ... , while i\ is held fixed. We denote by
~q the lattice determined by these h,k and by vq(x,t) the solution of
153
5 Hyperbolic equations in higher dimensions
(3.33a, b) defined on ~q' The ~q form a monotone increasing sequence of
denumerable sets in IRn+l. Their union 0 is again denumerable. The
function v q is define~ on all sets ~q' for q' <;; q .. The same holds for the
difference quotients 8t,8 av q (where the operator 8t,8 a is formed with respect
to the lattice ~q)' These difference quotients are bounded uniformly on ~q'
for q' <;; q when i + lal <;; 2, and are uniformly Lipschitz, when i + lal <;; 1.
From the boundedness it follows that there exists a subsequence S of
natural numbers q such that
liri:t 8~8avq(x,t)=ui,a(x,t)
(3.57)
qES
q-+oo
exists for i+lal <;; 1 and all (x,t)Eo. Moreover the ui,a are again uniformly
bounded and Lipschitz on o. Since the set 0 is dense in RT , we can
immediately extend the ui,a to all of the RT as Lipschitz continuous
functions with the same Lipschitz constant as had been found for the
8~8avq.
If we can prove that for (x, t) E RT and i + lal = 1
(3.58)
then the difference equation Avq=w will in the limit for q-+oo in S go
over into the differential equation Lu = w for u = uO,o, and we have solved
our initial-value problem (3.Ia, b). Indeed rewriting (3.33a) with the help of
(3.47b) and observing that
q= E- 12- q8.2v q
8.(1E-1)v
J
J
J
J
ui,a(x,t)=8iD~O,O(x,t)
1
-2
1 )8.v q = 8.v q (1 + E.J
J
J
1
-2
12- q8.2v q
EJ
J
and using the uniform boundedness of 8/v q immediately yields the transition from (3.33a) to (3.Ia).
We prove (3.58) for i = 1, a =O. The argument is the same for i = 0,
lal=1. Consider at first two fixed points (x,t) and (x,t+c) of o. There
exists then a q' such that (x,t) and (x,t+c) belong to ~q for all q;;. q'.
Prescribe an e > O. We can find a q" > q' such that
lu(x,t) - v q(x,t)1 < e,
lu(x,t+ c) - v q(x,t+ c)1 < e
for all q> q" belonging to S. (Here u stands for uo,o.) Thus
u(x,t) _ vq(x,t+c)-vq(x,t) 1< 2e
Iu(x,t+ c)c
c
c
for q> q", q E S. Here c is a multiple of k = A2 -q, say c = mk. Then
I
I
vq(x,t+c)-vq(x,t)
11 m-l
c
-8ov q(x,t) = m ,,~o 8ov q(x,t+vk)-8 ov q(x,t)
=I! ~l "~l
,,=0 ,.=0
<;;Mmk=Mc,
154
8Jv q (X,t+pk)1
I
3 Symmetric Hyperbolic Systems
where M is an upper bound for the second difference quotients of v q •
Hence
12e
- - - - - - 6ovq(x,t) .s;;;-+Mc.
I-U(X,t+C)-U(X,t)
c
c
Letting first q tend to
00
in S and then e tend to 0 we find that
I
- - - - - - u ' (x,t) .s;;;Mc
I-u(x,t+c)-u(~,t)
c
i 0
whenever (x,/) and (x,t+c) belong to o. By continuity of u and ui,o this
inequality holds then for any (x,t),(x,t+c) in R T • Letting here c tend to 0
yields the desired relation
ru(X,t) = ui,O(x,t).
This completes the existence proof for a solution u(x, I) of (3.1a, b) for
where T is sufficiently small, under the assumption that wE
C n + 2(R T ) and that w has compact support with respect to x.*
o.s;;; t .s;;; T,
PROBLEMS
1. Show that for the solution U of (3.1a, b) constructed here the domain of
dependence of u(x,t) on w(x,t) is contained in the pyramid
n
I
L IYr xjl.;;; X(t- s)
0.;;; s.;;; t.
(3.59)
j=i
2. Show that (3.1a, b) has a solution u(x,t) of class CS(RT) for sufficiently small T,
if w is of class cs+n+i(RT ) and of compact support in x.
3. Let u be a solution of O.la, b) of class C n + 2(RT) and of compact support in x.
Let v(x,t) be the solution of (3.33a, b) defined on the lattice r. Show that
u(x,t)- v(x,t)= O(h) for (x,t) on r, provided T and k/ h are sufficiently small.
(This implies that for w sufficiently regular we have vC~u for q~oo in any
maimer, not just for q in a "suitable" subsequence S. Thus, in principle, we have
a "construction" for u, a way to' find u numerically.) [Hint: Show that
8j U-Dj u=hfo'0dl} fo'drD/u(x+Orh)
forj=l, ... ,n
(3.60)
with a similar formula for 80 u - Dou. This implies that Au - Av = h W, where W
is of class C(RT ) and has compact support in x. Apply (3.56).]
-The fact that the domain of dependence of u(x,t) on w is finite (see problem 3, p. 145 and
problem I below) shows that the assumption of compact support for w is unessential.
155
6
Higher-order elliptic equations
with constant coefficients*
Here we can only indicate how some of the notions developed for the
Laplace equation apply to more general elliptic equations. We shall restrict
ourselves to a single linear homogeneous mth-order equation with constant
real coefficients for a scalar function u(x)=u(x1, ... ,xn). We write the
equation in the familiar form
P(D)u=O,
(0.1)
P(D)= ~ AaDa
lal=m
(0.2)
where
is an mth-degree form in D =(D 1, ... ,Dn) with constant real coefficients Aa'
Equation (0.1) is elliptic (see p. 58), if there are no real characteristic
surfaces, that is, if
(0.3).
P(~)*O for all real ~*O.
Weare only interested in the case when the dimension n is at least 2 and
when the form P does not vanish identically. In that case the order m OJ
the elliptic equation must be even. For taking any real vector r with
p(n*O we can find a vector 1/ in ~n which is independent of r. Then
P(q +
is an mth-degree polynomial in t with leading coefficient p(n*
o. Such a polynomial for odd m has at least one real root t, corresponding
to a real ~ = 1/ +
0 with P(~ = O. Accordingly we shall set m = 2/1 with a
positive integer /1.
The expression pm/I~lm is homogeneous of degree 0, does not vanish
on the sphere I~I = 1 and hence for all ~*O. Thus ellipticity implies the
tn
tr *
*([1]. [2]. [12]. [15]. [20]. [21])
156
I The Fundamental Solution for Odd n
existence of a c> 0 such that
(0.4)
For convenience we shall write the partial differential equation (0.1) as
Lu=( -I)"P(D )u=( -I)" ~ AaD"u=O.
lal =2"
(0.5)
1. The Fundamental Solution for Odd n
A fundamental solution for L with pole y is a function F(x,y) satisfying
LF=8y
(1.1)
in the distribution sense (see p. 69). For odd dimensions n it is easy to
obtain such an F by the method of decomposition into plane waves
described in Chapter 5, Section 5(d). We start with the observation that by
(2.81) of Chapter 5 the function
k(x,y)=dnr=dnix-yi
(1.2)
is a fundamental solution for the operator !l(n+ 1)/2:
f (!l~n+1)/2V(X)k(x,y )dx = v(y)
for every "test function" v E Cooo(\Rn). If then G(x,y) is a solution of
LG(x,y)=r=ix-yi
which has integrable derivatives of orders .;;;; n + m, we have in
F(x,y) = dn!l~n+ 1)/2G(X,y)
(1.3)
(1.4)
a solution of (1.1). For if v is a test function, and i is the operator adjoint
to L, we have by Green's identity
f
f (Lv )Fdx= dn (Lv)!l~n+1)/2Gdx
= dnf (!l~n+1)/2 Lv )Gdx
= dnf (i!l~n+1)/2v )Gdx= dnf (!l~n+1)/2v )(LG)dx
= f (!l~n+ 1)/2V )dnrdx = v(y).
Ident~ty
r(x,y,~
(2.79c) of Chapter 5 suggests the construction first of a solution
of
Lr=i(x-Y)·~i·
(1.5)
For then
(1.6)
157
6 Higher-order elliptic equations with constant coefficients
will be a solution of (1.3). Trivially
_I(x - y )·~121'+ I
f - (2JL+I)!P(~)
satisfies (1.5), leading to
--II
l(x-y)·~121'+1
(2 + 1)!P(~) dSf..
If-I= 1 JL
G(x,y)- cn
Setting x - y
(1.7)
(1.8)
= rq with 1711 = 1, we have
171·~121'+1 dS
(1.9)
(2 +1)!P(~) f..
I~I= 1 JL
Clearly the integral in (1.9) is of class C 21'+ 1 in the vector 71. Actually it
is C CXJ, even real analytic, in 71 for 1711 = 1. This can be seen by applying a
suitable orthogonal transformation to the variable of integration ~. Choose
a fixed unit vector ~. Let 71 be any unit vector linearly independent of ~.
Then 71 and ~ span a 2-dimensional subspace 'IT of IRn. There exists a unique
orthogonal transformation T (depending on 71 and of IRn which leaves all
vectors orthogonal to 'IT fixed, rotates'lT in itself, and takes ~ into 71. Clearly
C is determined by algebraic conditions, and must be analytic in 71 for real
unit vectors 71~ ±~. (For an explicit expression for T see problem 1
below.) We replace ~ in (1.9) by T~, which changes neither the domain of
integration nor the element of surface dSf.. Since
G(
x,y
)- -121'+11
-Cn
r
n
71·T~= n·T~= ~.~,
it follows that
G(
)-
x,y -
-121'+11
1~·~121'+1
dS
r
(2 + 1)!P(Tt) f..
If-I = 1 JL
s
Cn
Since T is real analytic in 71 for unit vectors 71 ~ ± ~, the same follows for
the integral. Since ~ was an arbitrary unit vector, we see that G is analytic
in 71 for all real unit vectors 71=(x-y)/lx-yl. Thus the only singularities
of G(x,y) for real arguments occur for r=O, that is, x=y.
By (1.8) G(x,y) is homogeneous of degree 2JL+ I in x-yo A derivative
of G of order k will be homogeneous of degree 2JL + 1 - k, hence integrable
for k <. 2JL + n = m + n. In particular
F(xy)=d c- 1a(n+I)/2
,
n n
x
r
J~
l(x-y)·~121'+1
(2JL+ I)!P(~)
dS
(1.10)
f.
If-I= 1
is homogeneous of degree 2JL - n = m - n, and hence is of the form
for x-y=r71,
(1.11)
where cf>(71) depends analytically on the vector 71 for real 71 with 1711 near 1.
We can draw up to JL of the Laplace operators in (1.10) under the integral
F(x-y)=r m - n c/>(71)
158
1 The Fundamental Solution for Odd n
sign, and obtain
(1.12a)
for n<m, and
I(x-Y)'~I
F(xy)=d c- 1Ll<n+l-m)/21
,
n n
P(~)
x
dS
I;
(l.12b)
1~1=1
for n>m.
When the number n of dimensions is even, the analogue of formula
(2.79c) of Chapter 5 for decomposition of functions into plane waves
becomes more complicated. The result that a fundamental solution becomes singular like r m- n need not hold. Terms of the order rm-nlogr may
occur. The simplest example is the fundamental solution (2'IT)-llogr for
the Laplace equation in two dimensions.
PROBLEMS
t. Prove that the orthogonal transformation T described on p. 158 is given by
T"=t._ a + b ,,+(1+2c)b-a 11
.. .. 1+ c ~
1+ c
."
(1.l3a)
b=~·t
(1.13b)
where
(Observe that T actually stays analytic in 11 for
1j =
r, that is, c = 1.)
2. Let the Aa in (0.2) stand for square matrices with n rows and columns, so that
(0.1) is an mth-order system for a vector u with n components. A fundamental
solution matrix F satisfies LF= 8y I where I is the unit matrix. Prove that for
odd n> m formula (1.l2b) represents a fundamental solution matrix if in the
integrand we write P -1 instead of 1/ P.
3. Use problem 2 to find a fundamental solution for the equations of elastic
equilibrium
O=/L~ui+(A+/L)
a:. (divu)
(1.14)
I
(see (2.7) of Chapter 1). [Hint: Involved is the computation of the integrals
(1.15)
for 1111 = 1. Introduce
This is a quadratic form in r. Show that for
implies that
Irl = I Q only depends on lI·r, which
159
6 Higher-order elliptic equations with constant coefficients
for 1'111 = I, with constants a, b. Evaluate Q for '11 = unit vector in the x3-direction.].
4. Find a fundamental solution of (1.l4) by finding a solution of the equations
JL.1u;+(A+ JL)
a:. (div)u)=F;(x)
(1.l6)
I
for F; E ColX>, using Poisson's formula, (1.28) of Chapter 4, and the equations
(A + 2JL).1(divu) = div F, .1r=2/r. [Answer: The fundamental solution matrix has
the elements
5. Show that solutions of the elliptic equation (0.5) are real analytic in the interior
of their domain of definition. [Hint: For odd n use the analyticity of the
fundamental solution. For even n use Hadamard's method of descent adding a
term ~:rl to P(g).]
2. The Dirichlet Problem
The Dirichlet problem for the equation (0.5) of order m = 2p. consists in
finding a solution u in the bounded region ~ for prescribed values of u and
its first p. - I normal derivatives on a~, or equivalently for consistently
prescribed values of u and its derivatives of orders ~ p.-l on a~. We can
instead solve an equation of the form
Lu=w(x) forxE~
(2.1)
with w prescribed in ~, where u satisfies the homogeneous Dirichlet
conditions
D~=O
for xEa~ and
lal ~ p.-l.
(2.2)
We shall show how at least a weak solution of this problem can be
obtained by following the Hilbert-space approach used in Chapter 4,
Section 5. The method used here for homogeneous L with constant
coefficients can be generalized to more general linear elliptic operators
with variable coefficients. The key elements are the Girding inequality and
the Lax-Milgram lemma. The passage from weak to strong solutions will
not be attempted here.
We denote by COOO(Q) the set of functions u E C OO(Q), whose support is
contained in a compact subset of ~. Using as norm the expression
lIuli =
1~
glal< p.
IDaul2dx ,
(2.3)
we can complete COO(Q) into a Hilbert space that will be called HP.(~).
Completing the subspace COOO(Q) with respect to the same norm leads to a
160
2 The Dirichlet Problem
Hilbert space HI)(Q) which we shall consider as the proper set of functions
satisfying the Dirichlet condition (2.2) in the generalized sense. *
We need to define an analogue B(u,v) to the bilinear form (5.13) of
Chapter 4. For that purpose we decompose each multi-index a with
lal=2,u in a trivial (nonunique) way into a=f3+y, where 1f31=IYI=,u.
Writing
(2.4)
we have
Lu=( -I)"P(D )u=( -1)1'
L B/3,yD/3+yu=w
= ="
1,81 Iyl
(2.5)
with certain B/3,y' Replacing, if necessary B/3,y by i(B,B,y + By,,B)' (again
denoted by B,B,). we can bring about that
B,B,y = By,,B'
(2.5a)
For any solution u E C 2"(Q) of (2.5) and any v E COOO(Q) we find by
repeated integration by parts that
k
vwdx=B(v,u)
(2.6)
where B denotes the bilinear functional
B(v,u)=
IQI,8I=IYI=fL
L B,BjD,Bv)(DYu)dx.
(2.7)
It is clear that for u E C 2"(Q) the partial differential equation (2.5) can be
replaced by the requirement that (2.6) holds for all "test" functions
v E COOO(Q). Here by Cauchy's inequality
(2.8)
IB(v,u)l..;; Kllvllilull
with a suitable constant K. This inequality permits us to extend the domain
of B(v,u) to all v,u in Ht;(Q). We just take Cauchy sequences of v,u in
COOO(Q) with respect to the norm (2.3), and observe that the corresponding
values of B converge by (2.8). The extended B is then again bilinear and
satisfies (2.8). Similarly given wE C "(Q) the expression
cJ>(v) =
k
vwdx
(2.9)
represents a linear functional in v satisfying
1cJ>(v)l..;; Ilvllllwll·
(2.10)
This functional also can be extended to HI)(Q) and is bounded because of
(2.10). Our modified version of the Dirichlet problem is then to find a
* HC(O) can also be obtained by completing the set CC(Q) of functions in
CI'(Q) satisfying
(2.2).
161
6 Higher-order elliptic equations with constant coefficients
u E HC(fl.) such that
cf>(v)=B(v,u)
(2.11)
for all v E Ht(fl.). The existence of u follows from the representation
theorem on p. 96, if we can prove the existence of a positive constant k
such that
klluIl 2';;;B(u,u)
(2.12)
for all u E HC(fl.). For by (2.5a) the form B is symmetric:
B(v,u)= B(u,v).
(2.13)
Moreover B(u,u»O for u*O by (2.12). Thus VB(u,u) = II lui II can be
used as a norm on Ht(fl.). Because of (2.12), (2.8)
(2.14)
Thus the new norm is "equivalent" to the old one, in the sense that
boundedness of functionals is the same in both norms. This applies in
particular to the functional cf>(v), and the existence of u in (2.11) follows.
It remains to establish (2.12). This is achieved most easily by Fourier
transformation, making use of Parseval's identity «3.22) of Chapter 5)
between a function g(x) and its transform t(g). More generally we have for
two real-valued functions u, v in CoCXl(n)
4f u(x)v(x)dx= f[(U+V)2_(U-v)2]dx
= f[lu(~)+ v(~)12_lu(~) -
v(~)12] d~
=2 f (u;;3+ a13 )d~=4Re f u;;3d~.
(2.15)
It follows for u E CoCXl(n) from (2.7), «2.18) of Chapter 5), (2.5), (0.4) that
B(u,u)=
~
1.81=IYI=JL
=Re
=
~
Bp,yf(DPu)(DYu)dx
1.81=IYI=JL
Bp,Yf«(i~)Pu)((i~)Yu)d~
f IPI=lyl=JL
~ Bp,y~P+Ylu (~)12d~
=f
~ Aa~alu(~)12d~
lal=2JL
= f P(~)lu(~Wd~:> c fl~12JLlu(~)12dt
162
2 The Dirichlet Problem
Since
'5'.
1Pr=1'
1(i~).8uI2 ~ ~ 1~121'IuI2= q~121'1u12
IfJl=1'
with a certain constant C, we find that
B(u,u)~ ~f ~ 1(i~).8umI2d~
IfJl=1'
=~f ~ (D.8U(X»)2dx.
C
IfJl=1'
(2.16)
Assume that {2 is contained in the cube Ix;1 ~ a for i = I, ... , n. Applying
repeatedly Poincare's inequality «S.19a) of Chapter 4) we have for 0 ~ p ~
p.
~ f (D YU)2dx ~ ~ (2a)21'-2v f (Di- VDYu )2dx
IYI=v
IYI=/f
~(2a)21'-2v ~ f (D.8U)2 dx
IfJl=1'
and thus by (2.3), (2.16)
I'
lIu11 2 = /f~0 lyt,l'f(Dyu )2 dx
~
f
(2a)21'-2v
1'=0
~
IfJl=1'
f(D.8u )2dx ~-kl B(u,u),
(2.17)
where
(2.18)
This establishes the inequality (2.12) for u E COOO(Q). It holds then clearly
also in the completed space HC({2), thus proving the existence of a weak
solution u of the Dirichlet problem.
We give some indications for the analogous existence proof for more
general linear elliptic equations Lu = w of order 2p., with coefficients
depending on x. It is tlien natural to write the differential equation in the
form
Lu=
~
IfJ I< I'
11'1< I'
(_1)I.8IB D.8+yu
.8, Y
(2.19)
and to introduce the bilinear form
B(v,u)= f
~ (D.8B.8,y v )(DYu )dx.
1.81" I'
(2.20)
IYI< I'
163
6 Higher-order elliptic equations with constant coefficients
A weak solution of the Dirichlet problem can then again be defined as a
u E HC(Q) for which (2.11) holds for all v E Hl)(Q). A difficulty about
applying the representation theorem arises from the fact that now B(v,u)
cannot be expected to be symmetric in v and u; (symmetry for the special
case considered earlier arises from the fact that there the differential
operator L is formally selfadjoint). This difficulty disappears, if we apply
the Lax-Milgram lemma which assures us that in a Hilbert space Hl) any
bounded linear functional cf>(v) can be written as B(v,u) with the help of a
suitable u in the same space, provided the given bilinear form B satisfies
inequalities (2.8), (2.12) for some positive constants k,K. (For a proof see
[14], [11].)
A further obstacle arises now from the circumstance that (2.12) just
does not hold in general. If it did, the Dirichlet problem for the equation
b.u + u = 0 would always be solvable, which is not the case. (See Problem 1
below.) What can be proved under suitable regularity assumptions is a
weakened form of (2.12):
Garding's inequality. There exists a positive constant k such that
kllul1 2 ,,;; B(u,u)+ ~ u 2 dx
(2.21)
holds for all u E Hl)(Q).
The integral on the right-hand side generally cannot be omitted. As a
consequence there is no existence for the solution of the Dirichlet problem
without some further qualification. Instead one obtains with the help of
Garding's inequality a statement in the form of an alternative: Either there
exists a nontrivial solution of Lu = 0 with Dirichlet data 0, or there exists a
solution u of Lu = w with Dirichlet data 0 for every sufficiently regular w.
PROBLEMS
1. Let n = 3 and Q be the ball Ixl < 17. Show that solution u of
vanishing boundary values can only exist, if
~u + u =
w(x) with
(2.22)
(compare problem 2, p. 79).
2. Show that for odd n the weak solution u of (2.5) constructed, is a strict solution
in the region Q. [Hint: Usr the fundamental solution, as on p. 99.]
3. Consider a homogeneous system of equations with constant real coefficients of
the form
Lu=( -1)I'P(D )u=( -1)1'
L
JaJ=21'
AuDuu=w,
(2.23)
where u and w are vectors with N components and the Aa constant square
164
2 The Dirichlet Problem
matrices. Let L be strongly elliptic in the sense that there exists a positive c such
that
(2.24)
for all real n-vectors
~
and N-vectors 1/. Write L in the form (2.5) and define
B(v,u)=
f 1.81 ~Iyl
=
= JL
(DPvlBp,y(DYu)dx.
Show that the "strong" Giirding inequality (2.12) holds for u E COOO(O) with a
suitable positive k. [Hint: Show that
B(u,u)=Re
f uTp(~)Ctd~
].
4. Show that the system (1.14) is strongly elliptic in the sense of problem 3.
5. Show that if u E CIL(O) and u E Bg(n) then u satisfies (2.2). [Hint: For uk E
CO(n) the identities
(2.25)
hold for lal..;; JL and all vE COO(O). Approximating uE COO(O) by certain Uk E
COOO(O), in the sense of the norm (2.3) the identities (2.25) are preserved in the
limit.]
165
7
Parabolic equations
1. The Heat Equation *
(a) The initial-value problem
The equation of heat for a function u= u(xI, ... ,xn,t)= u(x,t) has the form
u/=k!!:..u
(1.1)
with a positive constant conductivity coefficient k. For n = 3 the equation is
satisfied by the temperature in a heat-conducting medium. For n = I it
holds for the temperature distribution in a heat-conducting insulated wire.
The same type of equation occurs in the description of diffusion processes.
Applying a suitable linear substitution on x, t we transform (1.1) into
u/ =!!:..u
(1.2)
which will be used in the discussion to follow.
Equation (1.2) is parabolic. A characteristic hypersurface cp(x,t)= tl{I(x)=O has to satisfy the degenerate quadratic condition (see (2.24) of
Chapter 3)
n
~ l{I; =0.
k=I
(1.3)
'
Thus the only characteristic surfaces are the planes t = const. Unlike the
usual equations in mechanics (including the wave equation), equation (1.1)
is not preserved when we replace t by - t. This indicates that the heat
equation describes "irreversible" processes and makes a distinction between past and future (the "arrow of time"). More generally, (1.2) is
preserved under linear substitutions x' = ax, t' = a 2t, the same ones that
* ([30))
166
1 The Heat Equation
leave the expression Ix1 2/ t invariant. Thus it is not surprising that the
combination Ix1 2/ t occurs frequently in connection with equation (1.2).
Important information is obtained by considering the exponential solutions
with constant A and ~ = (~I' ... , ~n). Substitution in (1.2) yields the rdation
iA= _1~12, and hence
u(X, t) =
eix·~-IH't
(1.4)
For each fixed t;;' 0 equation (1.4) describes a plane wave function,
constant on the planes x·~=const. with unit normal UI~I, and repeating
itself when x is replaced by x+2'1TUI~12. Thus the waves in (1.4) have wave
length
2'1T
L=W
(1.5)
and amplitude
(1.6)
The solutions (1.4) decay exponentially with time, except in the trivial case
u = 1= const.
We have tacitly assumed in (1.4) that the vector ~ is real, so that u for
fixed t> 0 is bounded uniformly for x E~n. If we only consider our
solutions in the half space XI ;;. 0, it is natural to require boundedness just
in that half space, which leads to the condition that ~2' ... , ~n should be real
and Im~1 ;;. O. For example for a real positive A
~ = 0,
(1.7a)
is such a solution when
~I = V - ill. = (- 1 + i)Vf .
(1.7b)
The corresponding "physical" real solution would be
(1.7c)
For n = 3 we can interpret v as the temperature below ground in a flat
earth represented by the half space XI >0. Here v has boundary values
cos"At for XI =0, which oscillate periodically with frequency A and amplitude 1. The resulting temperature at depth XI still oscillates with frequency A but with a phase lag v' (1 /2lI.) X I and with an amplitude
167
7 Parabolic equations
exp( -
V(>../2) xl) that decays exponentially with depth. Thus at the depth
Xl
=Vf
log2
(1.7d)
the amplitude will have decreased to ·1/2 its surface value. This "halfdepth" is inversely proportional to VX or proportional to Vii, where
P = 2'17 />.. is the time period of v. Thus yearly surface variations of
temperature can be expected to penetrate V365 times = 19 times as deep
as daily variations with the same amplitude.
The "pure" initial-value problem for the heat equation consists in
finding a solution u(x, I) of
(1.8a)
ut-au=o for xElR n, 1>0
forxElR n , 1=0,
u=f(x)
(1.8b)
where we require uEC 2 for xElR n, t>O, and uEC o for xEIR;n, t~O. A
formal solution is obtained immediately by Fourier transformation. Writing
f(x) = (2'IT)-n/2
f eix°o/(~)d~,
(1.9a)
we would expect on the basis of (1.4) that
u(x,/) = (2'IT)-n/2
satisfies (l.8a, b). Substituting
f eixo~-I~I¥(~)d~
(1.9b)
f
j(~) = (2'IT)-:n/2 e-iy°o/(Y)dy
from Fourier's formula (2.15) of Chapter 5 and interchanging the integrations leads to
u(x,/)=
f K(x,y,t)f(y)dy,
(1.10a)
where
(1.10b)
The integral for K is easily evaluated by completing the squares in the
exponent; introducing a new variable of integration 11 by
~=
we find that
168
i(x-y)
21
1
+ Vi
1/,
1 The Heat Equation
Using the well-known formula
f e- 11112 = (f_: e-i' cis
drj
r
= 7T n / 2,
(1.10c)
we arrive at
K(x,y, t) = (47Tt) -n/2e -lx-YI 2/4t.
(1.10d)
We shall verify directly that the u from (1.1Oa, d) satisfies (1.8a, b),
without trying to justify the' steps in the formal derivation. In this we
follow the model of the proof of Poisson's formula (Chapter 4, (3.9».
Theorem. Let f (x) be continuous and bounded for x ERn. Then
u(x,t)= f K(x,y,t)f(y)dy
= (47Tt) -n/2
f e-lx-yI2/4'l'(y) dy
(1.11)
belongs to Coo for x ERn, t > 0, and satisfies ut = du for t > O. Moreover u
has the initial values f, in the sense that when we extend u by u(x,O)=
f(x) to t=O, then u is continuous for xER n, t~O.
The proof follows from basic properties of the kernel K:
for xERn,yER n, t>O.
(a) K(x,y,t)EC oo
(b)
(~t -dx )K(x,y,t)=O
(c)
K(x,y,t»O
(d)
fK(x,y,t)dy=1
(e)
For any 8 >0 we have
for t>O
for t>O
forxERn,t>O
lim!
K(x,y,t)dy=O
t-->O ly-xl>6
(1.12a)
(1.12b)
(1.12c)
(1.12d)
(1.12e)
t>O
uniformly for x ERn.
Here (a), (c) are trivial from (1.10d), and so is (b) from (1.10b). Also, by
(1.10d) substituting y = x + (4t)I/21j
(
K(x,y,t)dy = 7T- n/ 2 (
e- ll1 l'd1j.
)ly-xl>6
)1111>6/\1'4 t
(1.13)
When 8=0 this implies (d) by (1.10c), and implies (e) for 8 >0.
Clearly these properties of K show that the u defined by (1.11) belongs
to Coo and satisfies Ut = du for t > O. To prove that the extended u is
continuous, at t=O we have to show that u(x,t)~f(f) for x~g, t~O. For
169
7 Parabolic equations
e>O we can find a 8 such that If(y)-f(OI<e for ly-~1<28. Let
M= sup If(y)l. Then for Ix-~1<8
lu(x,t)- f(~)1 = If K(x,y,t)(J(y) -
<I
<I
ly-xl<8
f(~»~1
K(x,y,t)lf(y)-fml~+
ly-EI<28
I
IY-xl>8
K(x,y,t)lf(y)-fml~
K(x,y,t)lf(y)- f(~)I~ +2M (
K(x,y,t)~
J1y - xl >8
< ef K(x,y,t)~+2M
I
IY- xl>8
K(x,y,t)~<2e
for t sufficiently small.
By the same type of argument one proves more generally that if f(x) is
measurable and satisfies an inequality
(1.14)
for all x with fixed constants a, M then formula (1.11) defines a solution
u(x,t) of Ut =Au of class COO for x ERn and 0< t< 1/4a. Here u(x,t)~
f(O for x~~ and t~O at every point ~ of continuity of f.
We point out some important features of the special solution (1.11), (not
the only one), of the initial-value problem (1.8a, b). We observe that u(x,t)
for t > 0 depends on the values of f at all points. Equivalently the values of
fnear one ~ a moment later affect the value of u(x,t) at all x, though only
imperceptibly at large distances. Thus effects here travel with infinite
speed, indicating some limitation on the strict applicability of the heat
equation to physical phenomena. We notice from (1.12c,d) that for
boundedf
u(x,t) « f K(x,y,t)~)( s~p f(z») =
s~p fez),
and more generally that u satisfies the "maximum principle"
inf fez)
z
< u(x,t) < sup
fez)
z
for x ER n, t >0.
(1.15)
Here for continuous bounded f the equals sign can hold only when f is
constant.
The function u in (1.11) belongs to C 00 for any t > 0, even if the initial
values f of u are only continuous, or even have jump discontinuities. More
is true. For bounded continuousfthe function u(x,t) can be continued as
an analytic function to all complex x, t with Ret> O. We only have to
replace Ix-yl2 in the exponen.t in formula (1.10d) for K by the algebraic
expression (x-y)·(x-y). Then for complex x=~+i'IJ, t=a+i'f', with
170
1 The Heat Equation
~,1'/,y,(1,T
real, K(x,y,t) is analytic in x,t for t7"O, and moreover for (1)0
!K(x,y, t)! = (471") -n/2( (12+ T2) -n/4 exp ( _ Re
=
(~- y + ~1'/). (~~y + i1'/) )
4 (1+ IT
(1 + :: f/4eITJI2/4<JK( ~+ ~1'/,y,(1+ :).
(1.16)
It follows the for bounded continuous f from (1.12d) that
f
!u(~+ i1'/,(1 + iT)! < !K(~+ i1'/,y,(1 + iT)! ~ sup !f(z)!
= (
zERn
2
I+~
(12
)n/4
eITJI2/40 sup !f(z)!
zERn
(1.17)
for (1 > O. More precisely for bounded continuous f the function u(x, t) and
its first derivatives are represented by absolutely convergent integrals for
complex x, t with Re t > O. The analyticity of u follows (see p. 53).
Thus after an infinitesimal lapse of time a temperature distribution u (at
least if represented by (1.11)) is perfectly smooth, though of course, by
continuity, it will approximate the changes in theinitial data. This smoothness of the future has as its counterpart that the past is likely to be
rougher, as we shall see more precisely below. Values u(x,O)= f(x) that are
not analytic in x cannot have originated at all by conduction from a
temperature distribution in the past.
Formula (1.11) represents only one out of infinitely many solutions of
the initial-value problem (I.Sa, b). The solution is not unique without
further conditions on u, as is shown by examples of solutions u E
coo(lRn+l) of ut =/1u, which vanish identically for t<O but not for t>O.*
Following Tychonoff we construct such u for the case n = 1. They are
obtained by formally solving the partial differential equation ut = uxx for
prescribed Cauchy data on the t-axis:
u=g(t),
ux=O for x=O.
(US)
Writing u as a power series
DO
U
= ~ l5j( t)xj
j=O
• Conditions on u that imply uniqueness for the initial-value problem can take the form of
prescribing the behavior of u(x,t) for large Ix!. as shown below. We also mention a result of
Widder (see [30]) that there is at most one solution u which is nonnegative for t > 0 and all x.
The assumption u > 0 is reasonable when u is the absolute temperature.
171
7 Parabolic equations
we find by substitution into
powers of x that
Ut
=
gl=O,
go=g,
Uxx
and comparison of coefficients of
g;=(j+2)(j+l)~+2'
This leads to the formal solutions
(1.19)
They will be actual solutions if the power series can be shown to converge
sufficiently well. Choose now for some real a> I the g( t) defined by
g(t)= {exp[ -0 t-a]
for t>O
for t"; O.
(1.20)
Convergence of the corresponding series (1.19) depends on estimates for
the gk(t). These follow from Cauchy's representation for derivatives of
analytic functions
g(k)(t)= k!. r exp[ -Z-a] dz.
2m Jr (z _ t)k+ 1
(1.21)
Here for a real t > 0 we choose for our path of integration a circle
Iz - tl = fJt with a fixed fJ between 0 and I, and define za for Rez >0 as the
principal value of that function, which is real >0 for real z>O. For z on r
z = t + fJte i'" = t{1 + fJe hf )
Re{ - Z-a) = - t-a Re{1 + fJehf)-a
with a real ",. Obviously we can choose fJ so small that
Re{1 + fJehf)-a>~
(1.22)
for all real '" and hence
Re{ - z-a)< - it-a,
Ig(k)(t)1 < ~ exp [ _It-a].
(fJtl
2
(1.23)
Since k!/(2k)! < 1/ k! we observe that for real t >0 and any complex x
Thus, by comparison, the series (1.19) for
172
U
converges for real t>O and
1 The Heat Equation
complex x, and, of course, trivially also for t < O. Formula (1.24) shows
that limr-.ou(x,t)=O, uniformly in x for bounded complex x. The series
(1.19) as a power series in x is majorised (see p. 61) by the power series for
U(x,t)= {
:x
P[
1 t 1- a )]
1 (X2
t
0 - 2"
for t>O
for t ~O.
Since U(x,t) is bounded uniformly for bounded complex x and all real t,
the series (1.19) converges uniformly in x,t for bounded x and real t, and
the same holds for the series obtained by term by term x-differentiations.
In particular the series
co
g(k)(t)
co
L
X2k - 2= L
k=2 (2k-2)!
k=O
g(k+l)(t)
(2k)!
X2k
converges uniformly. Since this series is also obtained by formal differentiation of u with respect to t we find that ut = uxx ' More generally the
relation (8/8t)kU =(8/8xfkU holds, which implies that uEcoo(~n+l). We
observe that u is an entire analytic function of x for any real t, but is not
analytic in t, since u(O, t) vanishes for t ~ 0 but not for t > O.
PROBLEMS
1. Use formula (Lll) for n=1 to prove Weierstrass's approximation theorem: A
function f(x) continuous on a closed interval [a,b] can be approximated
uniformly by polynomials. [Hint: Define f(x) = feb) for x> b, f(x) = f(a) for
x < a. Then u(x,t)~f(x) for t~O uniformly for a.;;; x.;;; b, since u is continuous
for t;;. O. Approximate K(x,y,t) by its truncated power series with respect to
x-y.]
2. Letf(x) have uniformly bounded derivatives of orders .;;; s. Show that the u(x, t)
given by (Lll) is of class C S for t;;. 0 and all x. [Hint: Show that
f
D~= KD"jdy.]
3. Let f(x) be continuous in ~n and satisfy (Ll4). Show that the u defined by
(Lll) is analytic in x, t for all complex x and complex t = (J + iT with
(1.25)
4. Let u.(s,t), ... ,u,,(s,t) be n solutions of Ut=Uss ' Prove that
(1.26)
satisfies
Ut
=
au.
173
7 Parabolic equations
5. Show that for n= 1 the solution of (1.8a, b) withf(x)= 1 for x>O,f(x)=O for
x <0 is given by
(1.27a)
where q,(s) is the "error function"
q,(s) =
_2_1seVii
tl dt.
(I.2Th)
0
6. Show that for f(x) continuous and of compact support we have limr-.oou(x, t) =
o uniformly in x for the u given by (1.11).
7. For n= lletf(x) be bounded, continuous, and positive for all real x.
(a) Show that for the u given by (1.11)
lu(g+i'l/,t)l..;; e'12 / 4tu(g,t)
(1.28)
for real g, '1/, t with t > 0; [Hint: (1.16).]
(b) Show that
el/ 2
luX<x,t)I..;;-- sup u(x+y,t)
V2i lyl.;;v2i
(1.29)
(x,y,t real, t>O). [Hint: Use Cauchy's expression for uAx,t) as an integral
of u over the circle of radius V2i and center x in the complex plane.] (This
gives a means to estimate the maximum possible age t of an observed heat
distribution u in terms of its maximum and its gradient, assuming that it has
been positive and bounded for a time t.)
8. Find all solutions u(x, t) of the one-dimensional heat equation
form
u=
Ut
= Uxx
of the
Jt fC~ ).
[Hint: f(z) has to satisfy a linear ordinary second-order equation, of which one
solution f(z) = e- z2 is known, from u=K(x,O,t). All others can then be found
by quadratures.]
(b) Maximum principle, uniqueness, and regularity
Let w denote an open bounded set of ~ n. For a fixed T
cylinder Q in Rn+ I with base wand height T:
Q= {(x,t)lxEw, 0< t<
> 0 we form
T}.
the
(1.30a)
The boundary aQ consists of two disjoint portions, a "lower" boundary
a'Q, and an "upper" one a"Q (see Figure 7.1):
a'Q= {(x,t)leither xE
aw, O~ t ~ Tor xEw, t=O}
a"Q= {(x,t)IXEw, t= T}.
(l.30b)
(1.30c)
As in the second-order elliptic case the maximum of a solution of the heat
equation in Q is taken on aQ; but a more subtle distinction between the
forward and backwards t-directions makes itself felt:
174
1 The Heat Equation
a"n
T
a'n
a'n
a'n
o
w
x
Figure 7.1
Theorem. Let u be continuous in ~ and ut' UX,Xk exist and be continuous in g
and satisfy ut -du ..;; O. Then
(1.31)
maxu= maxu.
n
PROOF.
a'~
Let at first Ut-du<O in g. Let g, for O<e< T denote the set
g.={(x,t)IXEW, O<t< T-e}.
Since UECO(~,) there exists a point (x,t)E~, with
u(x,t)= max u.
n.
If here (x, t) Eg. the n~cessary relations ut =0, du";; 0 would contradict
ut-du<O. If (x,t) Eiy'g, we would have
Ut >0,
du";;O
leading to the same contradiction. Thus (x, t) E a'g" and
maxu= maxu..;;maxu.
n.
(l'~.
a'~
Since every point of Q with t < T belongs to some g. and u is continuous in
g, (1.31) follows. Let next Ut -du ..;; 0 in g. Introduce
vex, t) = u(x, t) - kt
with a constant positive k. Then vt-dv=u(-du-k<O and
maxu= max(v+kt) ..;;maxv+kT= maxv+kT..;; maxu+kT.
n
n
For k~O we obtain (1.31).
n
a'~
a'~
D
175
7 Parabolic equations
The maximum principle immediately yields a uniqueness theorem
n
Theorem. Let u be continuous in a!!:.d ut ' UXiXk e:;cist and be continuous in n.
Then u is determined uniquely in n by the value of Ut - 6.u in n and of u
on a/no
For the proof it is sufficient to consider the case where Ut - 6.u = 0 in
and u = 0 on a/no Applying (1.31) to u and - u we find that
maxu= max ( -u)=O,
~
n
(1.32)
~
and hence that u = 0 in n.
We can extend the maximum principle and the uniqueness theorem to
the case where n is the "slab"
n= {(x,t)lx E IRn, 0< t< T},
(1.33)
if we assume that u satisfies a certain growth condition at infinity.
Theorem. Let u be continuous for x E IRn, 0 < t < T, and let ut , Ux,xk exist and
be continuous for x E IR n, 0 < t < T, and satisfy
ut -6.u < 0 for 0< t< T, xE IRn
(1.34a)
u(x,t)<MealxI2
(1.34b)
u(x,O)= f(x)
forO<t<T,xElRn
for xE IRn.
Then
u(x,t) <supf(z)
z
forO<t<T,xElRn.
(1.35)
It is clear that this theorem implies that the solution of the initial-value
problem
ut -6.u=O for 0< t< T
(1.36a)
u(x,O)=f(x)
(1.36b)
is unique provided we restrict ourselves to solutions satisfying
lu(x,t)1 < MealxI2
for 0< t < T.
(U6c)
This shows that for bounded continuous f formula (1.11) represents the
only bounded solution u of (1.8a, b). Obviously the Tychonoff solution
(1.19), (1.20) for which u(x,O)=O cannot satisfy an inequality of the type
(1.36c). By (1.24) it does satisfy such an inequality with the constant a
replaced by 1/0t.
PROOF OF THE THEOREM.
tion that
It is sufficient to show (1.35) under the assump-
4aT<1
176
(1.37a)
1 The Heat Equation
For we can always divide the interval 0, t , T into equal parts, each of
length T< 1/4a, and conclude successively for k=O, 1, ... ,T/T that
U(x,/), sup u(y, kT) ,supu(y,O)
y
y
for kT, 1 ,(k + I)T. Assume then (1.37a). We can find an e >0 such that
4a(T+e)< 1.
Given a fixed y we consider for constants p, > 0 the functions
(1.37b)
vl'(x, I) = u(x,/) - p,(4'IT(T+ e- 1»)-n/2 exp[ Ix - Yl2 /4(T+ e - I)]
=u(x,/)-pK(ix,iy,T+e-/)
(1.38)
defined for 0,1, T. Since K(x,y,/) as defined by (l.lOd), with Ix-yl2
replaced by (x-y)·(x-y), satisfies Kt=!J.K for any complex x,y,t with
1:;60, we find that
ata vI' - !J.vl' = U -!J.u ,
O.
(1.39)
0< t< T}
(1.40)
t
Consider the "circular" cylinder
Q= {(x,/)lIx- yl <p,
of radius p. Then by (1.31)
v..... (y,t) ,max
V ...
(j'{l
...
(1.41)
Here on the plane part of a'Q, since pK > 0,
VI'(x, 0) ,u(x,O), sup J(z).
z
On the curved part
(1.42a)
Ix - yl = p, 0,1, T of a'Q by (1.38), (1.34b), (1.37b)
vI' (x,t) = Me alxl2 - p,(4'IT(T+ e - t»)-n/2 exp[p2 /4(T+ e- I)]
, Me o (lyl+p)2 - p,( 4'IT ( T+ e») -n/2e P2 /4(T+e)
, supJ(z)
z
for all sufficiently large p. Thus
max
v"... 'supJ(z).
(j'{l
It follows from (1.41), (1.38) that
vl'(y,t) = u(y,/) - p,(4'IT( T+ e - I») -n/2, supJ(z)
For p,~0 we obtain (1.35).
o
In order to derive regularity properties of a solution of the heat equation
in a bounded region we make use of Green's identity, as was done for
harmonic functions on p. 76. Let again Q denote the cylindrical region
(1.30a), where w is a bounded open set in IRn with sufficiently regular
177
7 Parabolic equations
n
boundary. Let u, up UXiXk exist and be con.!.inuous in and satisfy Ut - flu =
O. For an arbitrary function v(x, t) E C 2(n) we find by integration by parts
that
0= kv(ut -flu)dx
= - ( u(vt+flv)dx+ f vudx- f vudx
Jrl
xEw
xEw
t= T
t=O
- (Tdtf
(v du -udv)dSx •
Jo
xEilw
dn
dn
For a certain ~Ew and e>O we choose
(1.43)
v(x,t) = K(x,~, T+ e- t),
(1.44)
so that vt +tJ.v =0. Then for
f
xEw
vudx= f
t=T
e~O
xEw
K(x,~,e)u(x, T)dx~u(~, T),
since by the theorem of p. 169
f
(1.44a)
f
w(~,e)= K(~,x,e)u(x, T)dx= K(x,~,e)u(x, T)dx
is a solution of w.-flgw=O with initial values*
w(~,O)=u(x,T).
Since also K(x,~,T+e-t) is uniformly continuous in e,x,t for
xEaw, 0< t< T and for xEw, t=O, we find from (1.43) that
f K(x,~,T)u(x,O)dx
T f
(
du(x,t)
dK(X'~'T-t»)
+ i dt
d
-u(x,t)
d
dS
e~O,
u(x,T)=
w
o
xEilw
K(x,~,T-t)
n
x'
n
(1.45)
1)
We use this formula to extend u (~, T) to complex ~-arguments ~ = + iK
(with K real), keeping T real. The first integral in (1.45) trivially is an
entire analytic function of f Moreover for 0 < t < T, x =1=
1),
1)
K(x,~, T- t)= (4?T(T- t»-n/2 exp [ - (x-~)'(x-~)/4(T- t)]
is analytic in
~
and (see (1.16» bounded in absolute value by
(4?T(T- t»-n/2 exp [
IKI:(j~~)1)12l.
* The fact that x is integrated only over the region w instead over all of Rn does not change
the proof given on p. 169, as long as ~ is a fixed point of w.
178
I The Heat Equation
Thus K(x,~,T-t) is bounded uniformly for complex ~='I'/+i~ as long as
Ix -'111 2-I ~ 12 is bounded below by a positive constant. The same holds for
dK/ dn. In the second integral we first extend the t-integration from 0 to
T - e and then let e~O. Since sequences of analytic functions which
converge uniformly in a complex region have analytic limits, it follows that
u(~,t) is analytic in~, as long as IX-'I1f-I~12>0 for all xEaw. This is
certainly the case for complex ~ near a real point of w.
It follows that u(~, T) is real analytic for ~Ew. More precisely u(~, T) is
analytic for those complex ~ for which IIm~1 is less than the distance of
Re~ from aw. Hence a solution u(x,t) of ut-du=O is real analytic in x in
any open set where Ut and the UX,Xk are continuous. Moreover u(x, t) will be
an entire function of x if defined for all real x and for 1 restricted to an
open interval.
We easily conclude that solutions of ut-du=O in an open set QElRn+1
belong to C OO(Q). For if u has continuous x-derivatives of all orders, then
utxk = dUXk is continuous, and hence equals uxkt . Thus v = UXk also is a
solution of the heat equation with the same regularity properties as u. The
same holds again for VXk = UXkXk and then also for du = u(' Thus Utt = d~ is
continuous. Proceeding in this manner yields that all derivatives of u(x, I)
are continuous. As observed earlier analyticity of u(x,/) with respect to 1
cannot be expected. This fits in with the idea that the future of a heat
distribution does not depend exclusively on the past, but also on outside
influences that cannot be predicted.
PROBLEM
Let U be a solution of the one-dimensional heat equation Ut =
Uxx
in an open subset
n of the xt-plane. Show that at a point of n there exist constants A,M such that
I~;~ l";AM
k
(2k)!
for all nonnegative integers k. [Hint: Use that
U
(1.46)
is analytic in x.]
(c) A mixed problem
For n = 1 let u(x, I) be a solution of Ut - uxx = 0 in a half strip
O<x<L
0<1.
(1.47)
We seek the u satisfying the boundary conditions
u(O, I) = u(L,/) = 0 for I> 0
(1.48a)
u(x,O)=f(x)
(1.48b)
and initial condition
forO<x<L.
Here u might represent the temperature in an insulated rod with the ends
held at a constant temperature. This problem could be solved by Fourier
179
7 Parabolic equations
expansion with respect to x, as was done for the wave equation in Chapter
2. A solution in closed form is obtained by reflection. We continue f(x) to
all x so as to be an odd function of x and 2L - x:
f(x)=-f(-x),
f(x)=-f(2L-x).
(1.49)
We then solve the pure initial-value problem (1.Sa, b) with the extended*
function f by formula (1.11). The resulting u(x, t) satisfies u,- uxx = 0 and
the initial condition (1.4Sb). It also satisfies (1.4Sa) since u(x,t)+ u( - x,t)
and u(x, t) + u(2L - x, t) are again bounded solutions of the heat equation
with initial values 0, and hence vanish identically by the uniqueness
theorem.
Let cp(x) for x E IR be defined by
for 0 < x < L
for x < 0 or x > L.
Then the extended f satisfying (1.49) is given by
f( x)
</>(x)= { 0
00
f(x)=
L:
L:
~
n=
(</>(2nL+x)-</>(2nL-x).
(1.50)
-00
The corresponding solution (1.11) is
u(x,t)=
=
K(x,y,t)f(y)dy
K(x,y,t) n=~oo (</>(2nL+y)-</>(2nL-y)dy.
Substituting 2nL +y
u(x,t)=
f
oo
=~,
</>m
00
=
respectively 2nL - y
00
~
n=-oo
=~,
we obtain
(K(x,~-2nL,t)- K(x,2nL-~,t)d~
foL G (x,~,t)f(~)d~,
(1.51)
where by (UOd)
00
G(x,~,t)=
~
n=-oo
(K(x,~-2nL,t)-K(x,2nL-~,t)
= _1_ ~ (e-<x-~+2nL)2/4'_ e-<X+€-2nL)2/ 4t).
V4'ITt n = - 00
G can be expressed in terms of the classical theta-function
~3(Z,'T)=
1.
f
~ n=-oo
exp[ -i'IT(z+n)2/'T].
(1.52)
*The extended 1 generally will have harmless jump discontinuities at the points x=mL,
unless 1 satisfies the consistency conditions 1(0) = 1(L) =O.
ISO
2 The Initial-value Problem for Second-order Linear Parabolic Equations
We find
(1.53)
PROBLEMS
1. Find the solution of the one-dimensional heat equation u, = Uxx with boundary
conditions (1.48a,b) by expanding u into a Fourier sine series with respect to x.
Show that interchanging integrations we are led to the representation
u(x,/) =
fo
LG
(x,~, t)f(~)d~
with
Show that the identity between the expressions (1.53), (1.54) is equivalent to the
functional equation satisfied by the {l3-function:
_0.
"3
(Zr'--;I) =Vi
~ /T e i'1TZ2/T_o."3 ( Z,T.)
(1.55)
°
2. Find a solution u(x,/) of the one-dimensional heat equation u,-UXX=O in the
quadrant x > 0, I> satisfying the conditions
u(x,O) =0,
u(O,t)=h(/).
(1.56a)
[Hint: Apply formula (1.43) to the quadrant using
v=K(x,~,T+e-t)-K(x, -~,T+e-t).
(1.56b)
For e_O, using (1.44a) we arrive at
u(~, T)=
(T
~
Jo ~ (T- t)3/2
e-e/4(T-')h(/)dt.]
(1.56c)
2. The Initial-value Problem for General Second-order
Linear Parabolic Equations*
(a) The method of finite differences and the maximum principle
Restricting ourselves to one space variable x, we consider a linear equation
of the form
Lu= Ut - a(x, t)uxx - 2b(x, t)ux - c(x,t)u = d(x,t).
(2.1)
*([11], [18], [25D
181
7 Parabolic equations
We deal with solutions defined in a closed slab
(2.2)
J2={{x,t)lxER, o~ t~ T}
and satisfying the initial condition
(2.3)
u{x,O)=f{x).
To simplify statements we make the assumption that the coefficients a,b,c
belong to C OO(~) and that they and each of their partial derivatives are
bounded uniformly in ~. (Actually only a small finite number of derivatives will be required in each theorem). Most important is the additional
assumption (the "arrow of time"), that the coefficient a(x,t) is positive and
bounded away from in~:
0< inf a{x,t).
(2.4)
°
(X,t)E~
All results derived here will be based on the method of finite differences. The model is the discussion of symmetric hyperbolic systems in
Chapter 5, Section 3. However, the present situation is greatly simplified
by the existence of a maximum principle, which allows us to work with the
maximum norm rather than with L 2-norms.
Given two positive constants h, k we again consider the lattice ~
consisting of the points (x,t) with
x=nh,
t=mk
O~t~
T
(2.4a)
with integers n,m. We replace (2.1) by the difference equation
Av=
v{x,t+ k)- v{x,t)
v{x+ h,t) -2v{x,t)+v{x- h,t)
k
-a{x,t)
h2
-2b(x,t)
v(x+ h,t) - v(x- h,t)
2h
- c(x,t)v(x,t)=d{x,t)
for a function v(x, t) defined in the lattice
condition
v{x,O)= f(x)
~,
(2.5a)
and satisfying the initial
for x=nh.
(2.5b)
Solving for v(x, t + k) we write (2.5a) in the form
v(x,t+ k)= (Ail + hAb)v(x+ h,t) +(1-2Ail+ h2AC)V{X,t)
+ (Ail - hAb )v(x - h,t) + h2Ad(x, t),
(2.6)
where we have set
A=kjh2
(2.7)
and a,b,c are taken at the point (x,t). Introduce the norm
II gil = sup Ig(x)1
x
(2.8)
for a bounded function g (here defined for x = nh only). Introducing the
182
2 The Initial-value Problem for Second-order Linear Parabolic Equations
shift operators E,l1 defined by
Ev(x, t) = v(x + h, t),
."v(x,t) = v(x,t+ k)
(2.9)
a solution of (2.6) clearly satisfies
I11VI" [IAa+ hAbI +i1-2Aa+ ~2ACI +IAa- hAbI ]llvll + klldll. (2.10)
Assume now that A= k / h 2 is so small that the stability condition
2A sup a(x,t)<1
(2.11)
(x,t)E::£
is satisfied. Then since (2.4) holds and b,c are bounded,
IAa+"hAbI + 11-2Aa+h 2ACI +IAa-hAbI = I +h2Ac= I +kc
for all sufficiently small h. Setting
C=max(O, s~pc(x,t)),
(2.12)
II11Vll ,,(I + kC)lIvll + klldll,
(2.13)
we find from (2.10) that
where IIvll and IIdll depend on t=mk. Introduce the norm
IIldlll = sup IIdll = sup Id(x,t)l.
(x,t) E::£
t
(2.14)
Then iterating the inequality
II11Vll ,,(1 + kC)llvll + kllldlll,
and using the initial condition (2.5b), we arrive for t = mk at the estimate
IIvll ,,(I + kC)mllfll +
"emkcllfll +
b((1
+ kC)m-1)lIl d lll
.1
(e mkC -l)lIldlll
C
"eCtIIfII + teCtllldlll·
This proves:
Lemma I. If A= k / h 2 satisfies the stability condition (2.11) and if h is
sufficiently small, then the solution v of (2.5a,b) satisfies the "maximum
principle"
°
(2.15)
[This implies in the case d=O, c" that the supremum of Ivl in ~ is the
same as the supremum on the initial line.]
Let now u(x,t) be a solution of (2.1), (2.3), for which u,ut,ux,uxx are
uniformly bounded and uniformly continuous for (x,t)E~. Choose a fixed
A for which (2.11) holds. Consider the monotone increasing sequence of
183
7 Parabolic equations
lattices l:" corresponding to the choices h-2-", k_'J\2-2" for 11-1,2, ....
Denote by v" the solution of (2.5a,b) corresponding to l:". If U is the union
of all l:" and (x,t) a point of U, then v"(x,t) is well defined for all
sufficiently large 11. Under these assumptions we have
Lemma D. For each (x,t)E U
lim v"(x,t)-u(x,t).
(2.16)
" ..... 00
PROOF.
By Taylor's -theorem and the uniform continuity assumptions
u(x,t+k)-u(x,t)
k
-u,(x,t)
(2.17a)
u(x, +h,t)-u(x-h,t)
k
-uAx,t)
u(x+ h,t)-2u(x,t)+ u(x- h,t)
h
tend to 0 for
given e>O
h~O, k~O
2
-
(2.17b)
uxx(x,t)
(2. 17c)
uniformly for (x,t)ESl. This shows that for a
IAu-dl<e
for all (x, t) E l:" provided 11 is sufficiently large. (Here the difference
operator A is the one corresponding to the lattice l:".) Since v" satisfies
(2.5a) it follows that
IA(u-v")1 <e.
Using that u-v"-O for t=O, and applying (2.15) to u-v", we find that
lu(x,t) - v"(x,t)1 <;; eTe CT
(2.18)
for all (x, t) El:" provided 11 is sufficiently large. This proves Lemma II. 0
Since each v"(x,t) satisfies (2.15) for (x, t) El:" and
we see from (2.16) that also
lu(x,t)1 <;; e CT ( sup Iii + Tsup Idl)
11
sufficiently large,
(2.19)
n
for all (x,t)E U. The same inequality holds then for all (x,t)ESl because
of the assumed continuity of u.
We notice that the stability condition (2.11) requires the "mesh ratio"
k / h to tend to zero, as we refine the mesh and let h and k tend to O. This
requirement is to be expected from the Courant-Friedrichs-Lewy test (p.
6), if we want the solution of the difference scheme to approximate that of
the differential equation. Indeed bounded h / k would correspond to a
uniformly bounded domain of dependence for v"(x,t) oni, and hence also
for u(x,t), whereas the example of the heat equation shows that the
IR
184
2 The Initial-value Problem for Second-order Linear Parabolic Equations
domain of dependence of u(x,t) on the initial values is the whoie x-axis.
That the stability test involves a numerical bound on ak I h 2 is plausible
since this combination is dimensionless; [a(x,t) is of dimension x 2 1t by
virtue of the differential equation (2.1)].
We can dispense with the uniform continuity assumptions in (2.19):
Theorem. The inequality (2.19) holds for a solution u of (2.1), (2.3), if
U, Ut ' UX ' Uxx
are continuous and u, Ux are uniformly bounded in
~.
PROOF. We introduce a "cutoff function" cf>(x) ECCO(IR) for which 0, q,' I
for all x, while q,=0 for Ixl>2 and q,=1 for Ixl<l. Then for r>O the
function
u r (x, t) =q,(x I r)u(x, t)
agrees with u for Ixl < r and vanishes for Ixl >2r. ur,u;,u;,u;x are uniformly continuous and uniformly bounded in ~. Moreover u r satisfies
Lu r =q,d- 2ar- 1q,'ux
-
(ar- 2q," + 2br- 1q,')u= d*,
lur(x,O)I'lu(x,O)I=lf(x)l·
Thus by (2.19) applied to u r
lur(x,t)I' e CT ( sup If I+ Tsup Id*I).
IR
Q
(2.20)
Since u, Ux are bounded uniformly we have
lim sup Id*1 = sup Idl,
r---"oo
Q
Q
o
and (2.19) follows for u from (2.20).
The theorem trivially has the consequence that the solution of the
initial-value problem (2.1), (2.3) is unique, if we restrict ourselves to
solutions u for which u and Ux are bounded uniformly.
(b) Existence of solutions of the initial-value problem
Here we closely follow the pattern of the existence proof given in Chapter
5 for symmetric hyperbolic systems by the method of finite differences. We
only need to sketch the arguments. We assume that the prescribed functions f(x) and d(x,t) together with their derivatives of orders ,4 are
continuous and uniformly bounded for xEIR (respectively (x,t)E~).
Without ambiguity we shall now use the notation
IIfll= suplf(x)l,
IR
IIldlll= supld(x,t)1
Q
(2.21)
in the case of functions defined for all x (respectively all (x, t) E~). The
suprema of these functions in a lattice do not exceed these values.
Let again v denote a solution of (2.5a,b), where it is assumed that (2.11)
185
7 Parabolic equations
holds and that h is sufficiently small. We then have the estimate (2.15) for
vex, I) in ~. We can obtain analogous bounds for the difference quotients
of v. Let w(x, I) be defined in ~ by
I
w= 8v= h(E-I)v=
v(x+h,/)-v(x,/)
h
.
(2.22)
Using the product rule
8 (ab) =(Ea)8b+ (8a)b,
(2.23)
we find from (2.6) that w satisfies the equation
w(x, 1+ k) = E (Aa + hAb)w(x + h, I) + E (1- 2Aa + h2AC)W(X, I)
+ E (Aa- hAb)w(x- h,t)+ 8(Aa+ hAb)v(x+ h,/)
+ 8(1-&+ h2AC)V(X,t) + 8 (Aa- hAb)v(x- h,/)
+ kM(x,/)
= E (Aa+ hAb)w(x+ h,/)
+ E(I-2Aa+ h2AC+ hE- 1 8(Aa+ hAb»w(x,/)
+ E(Aa- hAb - hE -18 (Aa- hAb»w(x- h,/)
+k(8c)v+M).
Since 8a, 8b, 8c, are bounded uniformly by Illaxlll, IlIbxlll, lilexill respectively, we find for A satisfying (2.11) and h sufficiently small that
Iw(x,l+ k)1 <(1 + kEc+2k8b)lI w ll + k(18cllvl + IMI)
<(1 + kC + 2klllbxlll)lI w ll + k(lllexllllllvlll + Illdxlll)·
Since w(x, 0) =1, it follows as before that
1I18vlli = IIlwlll < e(c+21I1 bxlll)T (111xll + T(lllexllllllvlll + Illdxlll)),
(2.24)
where for Illvlll we have the estimate (2.15) in terms of 11111 and Illdlli.
Similar estimates clearly can be obtained for the higher difference
quotients 8 2v, 8 3v, 8 4v, and then also by (2.5a) for
I (
)
'Tv=7(1J- Iv =
v(x,l+k)-v(x,t)
k
'
(2.25')
and for 8'Tv, 8 2'Tv, and 'T2V. All these are bounded uniformly.
As before we define the increasing sequence ~P of lattices and the
corresponding solutions v P defined in ~p. Then v P, 8v P, 8 2v p , 8 3v p , 'TV P,
8'T2V Pare defined and bounded uniformly in ~P' and hence also in ~I' for
IL < P. For a suitable subsequence of the integers p
lim vP(x,t)=u(x,/),
11--+00
lim 8 2v p (x,t)=U"(x,/),
JI--+ 00
186
lim 8v P(X,/)=U'(x,/),
11--+00
lim'TVP(x,t)=u(x,/)
P-HX)
2 The Initial-value Problem for Second-order Linear Parabolic Equations
exist for all (x, t) in the union U of the ~p. Let (x, t), (y, t) be two points of
U, and hence of ~P' for all sufficiently large P. Then for O<y-x=hn=
n2- P
I
- - - - - -8v (x,t)
I-vP(y,t)-vP(x,t)
y-x
P
=lvP(x+nh,t)-vP(X,t) _ vP(x+h,t)-vP(x,t)
nh
h
=I(
I
E:;l - E;1 )vpl
=! (E~ll (En-2+2En-3+3En-4+ ... +n-l)V P
!
= 1~(En-2+2En-3+ .. . + n-l)~2vpl'
,
en ~ l)h 111~2VPIII
ly~xlll182vPIII ,Kly-xl
with a constant K independent of v. In the limit for
subsequence we obtain the inequality
I
- - - - - u ' ( x , t ) 'Ciy-xl
I-U(y,t)-U(X,t)
y-x
V-H~)
in the
(2.26)
for all (x,t), (y,t) in U. Since v P, 8v P, 8 2v p, 'TV are uniformly Lipschitz in
~P' the limits u,u',u",it are uniformly Lipschitz in U, and hence can be
extended as continuous functions to all of n. By continuity then (2.26)
holds for all (x,t), (y,t) in n. For y-7X we find that uxCx,t) exists and that
P
uxCx,t) = u'(x, t).
Similarly one verifies that Uxx = u", Ut = it, and that the differential equation
(2.1) is satisfied. This proves the theorem:
Theorem. If f, d and their derivatives of orders ,4 are continuous and
bounded uniformly, the initial-value problem (2.1), (2.3) has a solution
u(x,t), for which u,ut,ux,uxx are uniformly bounded and uniformly continuous in n.
In conclusion we observe that there are many results we derived for the
heat equation which one would expect to be valid for more general
equations of the form (2.1). For example the solution of (2.1), (2.3) should
be unique under the sole assumption that u, Up Ux' Uxx are continuous and
that u is bounded uniformly. We would also expect the solution of the
187
7 Parabolic equations
homogeneous equation Lu = 0 to exist and to be in Coo for 0 < t '" T, if f is
only assumed to be continuous and bounded (always requiring a,h,c to be
in Coo and bounded, and a to be positive and bounded away from zero).
However, such questions are beyond the scope of this volume.
PROBLEMS
1. Let u(x, f) be a solution of class C 2 of
U1 =
a(x,t)uxx + 2b(x,t)ux + c(x, f)u
in the rectangle
n= {(x,t)IO< x < L, 0 < t < T}.
Let a'n denote the "lower boundary" of n consisting of the three segments
x=O,
0< t< T
O<x<L,
t=O
x=L,
0< t< T.
(a) Prove that in case c<O in n
lu(x,t)1 < sup lui for (x,t) En.
a'Sl
[Hint: Show maxgu cannot be assumed on
(b) Show that more generally
n- a'n unless maxgU < 0.]
(2.27)
where
C=max(O, maxiel).
Sl
[Hint: Substitute u= e'Ylv, where y> C, and apply part (a) to v.]
2. Let u(x,f) denote the solution (1.11) of
Lu=Ut- uxx=O for 0< t < T
(2.28a)
u(x,O) = f(x),
(2.28b)
where f is continuous. Let ~ denote the lattice of points (x,t) with x,t of the
form x = nh, t = mk and v the solution of
v(x,t+ k)-v(x,t) v(x+ h,t)-2v(x,t)+ v(x- h,t)
=0 (2.29)
k
h2
with v(x,O)= f(x).
Av=
(a) Show that for A=k/h 2 =1/2
v(nh,mk)=2-m
f
j=O
(~)f«n-m+2j)h)
J
and hence
sup Ivl < sup IfI
~
188
III
(2.30)
2 The Initial-value Problem for Second-order Linear Parabolic Equations
(b) Show that for a fixedfECo"'(IR), (kA';; 1/2 and (x,t)E~
lu(x,t)-v(x,t)1 =O(h4).
Show that for the special value A= 1/6 the better estimate
lu(x,t)-v(x,t)1 =O(h6)
holds. [Hint: Expand Au by Taylor's theorem, using u/ = uxx .]
(c) Let O<A';; 1/2 and ~Y be the lattice corresponding to h=2- Y, k=A2- 2Y,
and let v Y be the corresponding solution of (2.29) with initial valuesf. Let U
be the union of the ~Y for v = 1,2,3, .... Show that for (x, t) E U
lim vY(x,t)=u(x,t)
p--->oo
(2.31)
provided f has bounded continuous derivatives of orders .;; 4. [Hint: Use the
theorem on p. 187 and known properties of u.]
(d) Show that (2.31) holds assuming only that O<A';; 1/2 and that f(x) is
continuous and has compact support. [Hint: Approximate f uniformly by
functions with bounded derivatives of orders .;; 4; use the maximum principle for the v'.]
189
Bibliography
[1] Agmon, Shmuel, Lectures on elliptic boundary value problems, Van
Nostrand, 1965.
[2] Bers, Lipman, John, F., Schechter, M., Partial Differential Equations, Interscience Publishers, 1964.
[3] Bremermann, Hans, Distributions, Complex Variables and Fourier Transforms, Addison Wesley Publishers, 1965.
[4] Carrier, G. F., and Pearson, C. E., Partial Differential Equations, Academic
Press, 1976.
[5] Carroll, R c., Abstract Methods in Partial Differential Equations, Harper
and Row, 1969.
[6] Courant, R, and Friedrichs, K. 0., Supersonic Flow and Shock Waves, 1948,
(Repr.), Springer-Verlag.
[7] Courant, R., and Hilbert, D., Methods of Mathematical Physics, Interscience
Publishers, Vol. I, 1953; Vol. II, 1962.
[8] Dunford, N. and Schwartz, J. T., Linear Operators, Part II, Interscience
Publishers, 1963.
[9] Epstein, B., Partial Differential Equations, McGraw Hill, 1962.
[10] Fichera, Gaetano, Linear Elliptic Differential Systems and Eigenvalue Problems, Lecture Notes in Mathematics, 8, Springer-Verlag, 1965.
[11] Friedman, Avner, Partial Differential Equations of Parabolic Type, Prentice
Hall, 1964.
[I2] Friedman, Avner, Partial Differential Equations, Holt, Rinehart and Winston, 1969.
[13] Garabedian, P. R, Partial Differential Equations, John Wiley & Sons, Inc.,
1964.
191
Bibliography
[14]
Hellwig, G., Partial Differential Equations, New York, Blaisdell, 1964.
[15]
Hormander, Lars, Linear Partial Differential Operators, Springer-Verlag,
1963.
[16]
Isaacson, E., and Keller, H. B., Analysis of Numerical Methods, John Wiley
& Sons, Inc., 1966.
[17] John, Fritz, Plane Waves and Spherical Means Applied to Partial Differential
Equations, Interscience Publishers, 1955.
[18] John, Fritz, Lectures on Advanced Numerical Methods, Gordon and Breach,
1967.
[19]
Lax, P. D., Lectures on Hyperbolic Partial Differential Equations, Stanford
University, 1963.
[20]
Lions, J. L., Equations differentieles operationelles et problemes aux limites,
Springer, 1961.
[21]
Lions, J. L., and Magenes, E., Non-homogeneous boundary value problems,
Springer-Verlag, 1972.
[22]
Mikhlin, S. G., Linear Equations of Mathematical Physics, Holt; Rinehart
and Winston, 1967.
[23]
Petrovsky, I. G., Lectures on Partial Differential Equations, Interscience
Publishers, 1954.
[24]
Protter, M. H. and Weinberger, H. F., Maximum Principles in Differential
Equations, Prentice Hall, 1967.
[25]
Richtmyer, R. D. and Morton, K. W., Difference Methods for Initial Value
Problems, 2nd Ed., Interscience Publishers, 1967.
[26]
Smimov, V. I., A Course of Higher Mathematics, Vol. IV, Translation,
Addison Wesley, 1964.
[27] Tikhonov, A. N. and Arsenin, V. Y., Solution of Ill-Posed Problems, Winston/Wiley, 1977.
[28]
Tikhonov, A. N. and Samarskii, A. A., Equations of Mathematical Physics,
Translation, Pergamon Press, 1963.
[29] Treves, F., Basic Linear Partial Differential Equations, Academic Press, 1975.
[30] Widder, D. V., The Heat Equation, Academic Press, 1975.
192
Glossary
IR = set of real numbers
lIl"=real n-dimensional space (usually endowed with euclidean metric)
n c IRn define:
n= closure of n
For a set
an = boundary of n
ninl = interior of n
wen is compact in n
if
wis a closed and bounded subset of ninl.
For a function j with domain nclRn define: suppj=support of j=
closure of subset of n wherej=;eO.fhas compact support in n ifj=O outside
a closed and bounded subset of n inl (that is, if j vanishes in a neighborhood of an and outside some ball.)
For
n c IRn define:
Co(n) = set of functions j continuous in n
CS(n) = set of j with continuous derivatives of orders
~s
in n
Co(n) n CS(ninl) = set of j continuous in the closure of n and with continuous derivatives or orders ~s in the interior.
CO(n) = set of j with continuous derivatives of orders
compact support in n.
~s
in
n
and of
For a matrix a = (aik)
aT = transpose of a = matrix b = (bik ) = (aki) obtained by interchanging rows
and columns of a.
a> 0, if aT = a and the quadratic form ~ aik~i~k is positive definite
i,k
O-notation: u= O(v) if there exists a real constant K such that
all pairs u, v under consideration.
lui ~Kv for
multiple-index notation: see pp.52- 53.
~ = Laplacian '2.
a2/ ax?
D = d'Alambertian = 0 2 /
at 2 -
~
193
Index
A priori estimates 143
Adjoint operator 65, 157
Adjoint problem 66
Amplitude 167
Analytic 173
Analytic function 2
Approximate solution 113, Il8
Arithmetic mean 77
Arrow of time 166, 182
Attraction 81
Autonomous 9
Autonomous system 21
Average 103
Cauchy problem 10, II, 12, 13, 14, 23,
26,29,31,32,54,59,60,75,76, Il2, 113,
132
ClIuchy-Riemann equations 2,80
Cauchy-Schwarz inequalty 149
Cauchy sequence
172
Backward characteristic 46
Backward characteristic cone 114
Backward difference quotients 7
Barrier functions 92
Barrier postulate ·92
Beltrami equation 36
Bessel function III
Biharmonic equation 3, 80
Binomial theorem 53
Blow-up 17
Bounded functional 95
Canonical form 46
Cauchy data 32, 33, 55, 59, 113
Cauchy for the quasi-linear equation
Cauchy formula 53
Cauchy inequalities 95
Cauchy integral 125
Cauchy-Kowalewski theorem 59
95
Cauchy theorem 80
Cauchy's representation for derivatives
10
Characteristic 32, 55, 58, 59
Characteristic, backward 46
Characteristic curves 9, 10, 13, 18,21,23,
33, 34, 36, 45, 59
Characteristic differential 45
Characteristic direction 9, 10, 19
Characteristic elements 24, 28
Characteristic equations 22
Characteristic form 56
Characteristic hypersurface 166
Characteristic line 4
Characteristic manifolds 52, 58
Characteristic matrix 56, 57
Characteristic projections 13, 59
Characteristic strip 22, 23, 24, 26, 28
Characteristic surfaces 55,57, 58
Characteristics 13, 31, 36, 39
Compact support 67
Compactness 87
Compatibility conditions 31,32,41, 55
Completeness 87, 95, 160
Conductivity coefficient 166
Conjugate harmonic 2,80,81
Connected 83
Conoid 29
Conservation law 17
195
Index
Consistency conditions 118, 119, 180
Contractive 47
Correctly-set problems 131
Courant-Friedrichs-Lewy test 6, 7, 8, 184
Courant-Lax theory 44
D'Alembertian 103
Darboux's equation 104
Data 11
Decay 109, 167
Descent 110
Difference equation 5, 182
Difference quotients 151
Differencing a product 151
Differentiation operator 53
Diffusion 166
Dirac function 68
Direction numbers 9, 19
Dirichlet integral 96, 102
Dirichlet norm 97
Dirichlet principle 102
Dirichlet problem 73, 81, 82, 84-102,
132, 160, 163, 164
Dirichlet problem, weak solution of 164
Discontinuity 34
Distribution 68, 70, 75
Distribution solutions 67, 69
Distributions 69
Disturbances 40
Divergence form 17
Divided difference operators 147
Domain of dependence 5,6, 39, 107, 108,
III, 113, 119, 155
Domain of influence 115
Duhamel's principle 112, 120
Eigenfunctions 42, 102, II 7
Eigenvalue 102, 117
Elastic
equilibrium 159
waves 3, 110, 133
Element(s) 22,24,26,28
characteristic 24
line 23
Elliptic equations 33, 36, 58, 83, 156, 163
Energy 44
estimates 151
identity 73, 140
inequality 139, 141
integral 116, 119
norm 107
Envelopes 20, 28
Euler-Lagrange equations 30
Euler-Poisson-Dabroux equation 105
Euler's P.D.E. for homogeneous functions
15
Extremal property 89
Extremals 30
196
Finite differences, method of 146, 181
First-order
equation 1
systems 44, 139
Flow 2
Focussing 106, 109
Form
canonical 46
characteristic 56
divergence 17
normal 36
standard 60
Fourier
formula 168
transform 122
transformation 122, 162, 168
Functional 67
bounded 96
linear 95
Fundamental solution(s) 70, 74, 75, 79,
80, 84, 137, 157, 159, 160, 164
for the Laplace equation 159
with pole 69
Garding's
hyperbolicity condition 125, 128, 133,
145
inequality 160, 164
strong inequality 165
Gauss divergence theorem 64
Gauss's law of the arithmetic mean 77
General solution 10
Generalized solutions 33, 40, 67
Geometric optics 26
Gradient catastrophe 17
Gradient 52
Gravitational attraction 80
Green's
function 84, 88
identities 72, 142
Hadamard's method of descent 110
Hamilton-Jacobi equation 30
Harmonic 75, 76, 77, 83, 87, 88
conjugate 2, 80, 81
functions 2, 75, 76, 86
Harnack's inequality 88
Heat
conduction 3
distribution 174
equation 3, 166, 168, 174, 177, 179
Hilbert space 94,95, WI, 160
Hodograph method 37
Holmgren's theorem 66, 129, 146
Homogeneous
function 15
system of equations 164
Index
Huygens's
construction 30, 107
principle in the strong form 108, 111
Hyperbolic 33, 36, 38, 45, 122
equations 103, 120
second-order equation 139
strictly 129, 136
symmetric 145
Hyperbolicity 58, 128, 129
Improperly posed problems 131
Incorrectly set problems 131
Influence 5, 40, 107
Initial
boundary value problems 40, 116
conditions 120
curve 23
data 139
functions 39
value problem 11, 39, 45, 103, 120,
145, 166, 170, 185
value problem for the wave equation
106
values 4, 11
velocity distribution 16
Inner product space 95
Instability 6
Integral 22
surface 19, 23, 24, 28, 29
Integration by parts 64
Irreversible 166
Jump 17, 34, 35
condition 18
intensity 35
Korteweg-deVries equation 4
Lagrange-Green identity 64, 65, 67, 69
Lame constants 3
Laplace
equation 2,72,73,75,76,78, 159
operator 2, 58, 65, 72
Lattice 153, 182
Lax-Milgram le=a 160, 164
Legendre transformation 37
Level lines 27
Line element 23
Linear
equation 1, 13, 54
first-order system 57
functional 95
parabolic equations 181
second-order equation 35
Liouville's theorem 88
Lorentz transformation 116
Lower boundary 174, 188
Majorise 61,62,63, 173
Maximum
norm 47,182
principle 81, 82, 83, 88, 170, 174, 181,
183
Maxwell's equation 2
Mean value property 77,79
Mesh ratio 7
Minimal surface 3, 37
Mixed initial-boundary-value problems
48, 119, 133, 179
Modified Dirichlet problem 94
Monge cone 19,20,26,29
Monomial 52
Multi-index 52
Navier-Stokes equations 4
Neumann problem 73, 81
Newton's law 80
Noncharacteristic 55, 56, 57, 113
Norm 160
equivalent 162
maximum 182
Normal
derivative 32
distance 28
form 36
Operator
adjoint 65, 157
divided difference 147
Laplace 2, 58, 65, 72
shift 6, 147
Order 1
O.D.E. 9
Pailey-Wiener theorem 129
Parabolic 33, 36
Parallel curves 28
Parseval's identity 144, 162
Perron's method 89
Phase lag 167
Plane
element 22
wave function 135, 136, 157, 159
Poincare inequality 97, 102, 163
Poisson's
differential equation 78
formula 77, 88, 99, 100, 160, 169
integral formula 84, 85, 88
kernel 85
Polytropic gas 3
Potential 81
functions 2
theory 73
velocity 2, 3
Principal part 56, 57, 128
Propagating waves 38
197
Index
Propagation 23
of discontinuity 35
of singularities 33, 58
speed of . 131
Pure initial-value problem 43, 48
Quasi-linear
equations 1, 8, 18
systems 48, 50
Radon-problem 138
transformation 138
Ray 23,27
Real analytic 53, 160
Reduced wave operator 79
Reflection 76, 84, 88, 180
Regularity 75, 177
Representation theorem 96
Round-off errors 6, 8
Schrodinger's wave equation 3
Schwartz notation 54
Second-order equation 31
Selfadjoint 75
Separation of variables 42
Shift operators 6, 147
Shock condition 17
Signals 40
Simple wave 50
Smooth 171
Sobolev inequalities 144, 146, 152
Solution
approximate 113, 118
distribution 69
fundamental 69, 70, 74, 75, 79, 80, 84,
137, 157, 159, 160
general 10
generalized 33
strict 34, 143
weak 17, 69, 40, 141
Space-like 27, 114, 115, 119, 122, 132,
. 146
Speed of propagation 18, 131
Spherical
means 103, 104
synunetry 73, 79, 80, 109
Stability 7, 147, 183
Standard
form of the initial-value problem 60,
120, 129, 131, 137
Strict solution 34, 143
Strictly hyperbolic 129, 136
Strip 26
characteristic 22, 23, 24, 26
condition 22, 23, 31
Strongly elliptic 165
Subharmonic 77, 79, 83, 89
Symbol 56, 57
198
Symmetric hyperbolic 139, 140, 145
System
autonomous 21
first order 139
of equations 164
of first order equations 44
quasi-linear 48
undetermined 21
Telegraph equation III
Temperature distribution 166, 167
Test function 67, 69, 157
Theta-function 180
Time-like 27, 115
Triangle inequalities 95
Tricomi equation 36
Tychonoff solution 171, 176
Type .33
Undetermined system 21
Unique continuation 53
Uniqueness 12
for the initial value problem
theorem of Holmgren 65
145, 176
Variational problem 30
Vector space 94
Velocity
field 16
potential 2, 3
Vibration of string 44
Water waves 2,4
Wave
fronts 26, 27
operator 79
propagating 5
Wave equation 38, 106, 121, 133, 138,
146
homogeneous 112
in 5 dimensions 109
in n-dimensional space 103, 127
in 3 dimensions 138
inhomogeneous 112
initial problem for the 132
standard problem for the 127
Wavelength 167
Waves
circular 30
elastic 3, 110
plane 135, 157, 159
propagating 38
simple 50
water 2,4
Weak solutions 17, 18, 40, 141, 160, 164
Weierstrass's approximation theorem 173
Well-posed problems 131, 132