Partial Differential Equations

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Lectures in

APPLIED MATHEMATICS Volume 3A

Partial Differential Equations

American Mathematical Society

LECTURES IN APPLIED MATHEMATICS Proceedings of the Summer Seminar, Boulder, Colorado, 1957

Volume I

Probability and Related Topics in Physical Sciences BY MARK KAC with G. E. UHLENBF.CK, A. R. HIBBS, and BALTH. VAN DER POL

Volume II

Fluid Mechanics By SYDNEY GOT DSTE.TN with J. M. BURGERS

Volume III

Partial Differential Equations By LIPMAN BERs, FRITL JOHN, and MARTIN SCHECHTER

with LARS GARDING and A. N. MILGRAM

Volume IV

Solid Mechanics By R. S. RIVI.IN with W. PRAGER

LECTURES IN APPLIED MATHEMATICS Proceedings of the Summer Seminar, Boulder, Colorado, 1957

AMERICAN MATHEMATICAL SOCIETY EDITORIAL COMMITTEE

ALSTON S. HOUSEHOLDER, Chairman Oak Ridge National Laboratory

MARK KAC Rockefeller University

H. J. GREENBERG International Business Machines Corporation

VOLUME III

PARTIAL DIFFERENTIAL EQUATIONS By

LIPMAN BERS

FRITZ JOHN MARTIN SCHECHTER Courant Institute of Mathematical Sciences, New York University

WITH SPECIAL LECTURES BY

Lars Girding University of Lund, Lund, Sweden

and the late A. N. Milgram University of Minnesota.

AMERICAN MATHEMATICAL SOCIETY

Providence, Rhode Island 02904

Copyright Q 1964 by John Wiley & Sons, Inc. All Rights Reserved This book or any part thereof may not be reproduced in any form without the written permission of the publisher. SECOND PRINTING, OCTOBER, 1966 BY JOHN WILEY & SONS, INC. THIRD PRINTING, 1971 BY THE AMERICAN MATHEMATICAL SOCIETY

International Standard Book Number 0-8218-0047-3 Library of Congress Catalog Card Number 63-19664

Printed in the United States of America

ARTHUR N ORTON MILGRAM

June 3, 1912-January 30, 1961

Arthur Norton Milgram died suddenly on January 30, 1961. He was a penetrating mathematician and an inspiring

teacher. The surviving authors, who were his friends, dedicate this volume to his memory.

Foreword

This is the third volume of the Proceedings of the Summer Seminar on Applied Mathematics, sponsored by the American Mathematical Society and held at the University of Colorado over the four weeks beginning June 23, 1957

The purpose of the Seminar was tutorial, to present to mature mathematicians the current status of the theory in several fields covered, and to pose for them some of the more pressing and interesting mathematical problems lying open in these fields. It could be described

as an endeavor to promote cooperation of mathematicians with theoretical physicists. The publication of these volumes is intended to extend the same information to a much wider public than was privi-

leged to actually attend the Seminar itself, while at the same time serving as a permanent reference for those who did attend. The program of the Seminar was organized by a committee of the American Mathematical Society with the following membership:

P. R. Garabedian A. S. Householder Mark Kac R. E. Langer

C. C. Lin Wm. Prager J. J. Stoker M. H. Martin, Chairman

Local arrangements, including the social and recreational program, were organized by a committee of the Department of Applied Mathematics, University of Colorado, as follows:

J. R. Britton R. Ben Kriegh L. W. Rutland

L. C. Snively

K. H. Stahl C. A. Hutchinson, Chairman

The indefatigable energy and enthusiasm of the chairmen, and the

cooperation of other members of the university staff, contributed

viii

FOREWORD

immeasurably to the successful execution of the plans for the seminar and to the enjoyment of the participants. The Seminar opened Sunday evening, June 23, with an address by Professor Richard P. Feynman, California Institute of Technology, on the subject "The Relation of Mathematics to Physics." The formal, technical sessions were held in the mornings, leaving the afternoons free for study and for holding informal discussion groups on related special topics. Several such groups were organized and met regularly throughout the period of the Seminar. A. S. HOUSEHOLDER

Preface

In June and July of 1957 we delivered lectures on the theory of partial differential equations at the Seminar on Applied Mathematics which the American Mathematical Society held at the University of

Colorado in Boulder. Mimeographed notes of these lectures were prepared at that time, and a printed version of the lectures was promised. We regret that pressure of other work delayed the publication for so long. The chapters on hyperbolic equations by F. John which appear here are an edited version of the original notes, in the preparation of which S. V. Parter gave valuable aid. The lecture notes on elliptic equations,

delivered by L. Bers, have been considerably expanded. Such an expansion would not have been accomplished had not M. Schechter agreed to assume co-authorship of this part of the book.

At the Seminar L. Girding and the late A. N. Milgram delivered two lectures, the text of which is included in this volume. No attempt has been made to achieve artificial unity of presentation

by avoiding all overlaps or by coordinating notation. The diversity of approaches, methods, and points of view is characteristic for the theory of partial differential equations. In the informal and stimulating

atmosphere of the Boulder Seminar this diversity became quite apparent; we hope that some of it is preserved in the printed version. The bibliography is far from complete; we list primarily the books and papers which we found helpful in preparing these lectures. This is neither a textbook for beginners, since most standard material is omitted, nor a reference book f o r specialists. I t is aimed, as the oral presentation was, at mathematicians not necessarily familiar with the field of partial differential equations who %%-ant to acquire an understanding of some of the problems and methods in this discipline. The technical prerequisites are very modest. A student familiar with the ix

x

PREFACE

fundamentals of real and complex function theory and with the elements of functional analysis should have no difficulty in following the presentation. LIPMAN BERS

FRITz JOHN

New York

March 1964

Acknowledgments

The authors are indebted to Anneli Lax for her help in rewriting the first part and to Alfred Schatz for the preparation of the index to that part. Concerning the second part, they would like to express their gratitude to Deborah Schechter for her help in preparing the manuscript, to Michael Herschorn and Joseph Ercolano for corrections, to Lesley M. Sibner for reading the proofs, and to Lynnel Marg for preparing the index. Appreciation is expressed to Helen Samoraj for expertly typing the entire manuscript. L. B.

F. J. M. S.

Contents Part I. Hyperbolic and ParaboUc Equations, by Farrz JoHN .

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1. Equations of Hyperbolic and Parabolic Types

2. The Wave Operator . . . . . . 2.1. The one-dimensional wave equation

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2.2. The initial value problem for the wave equation in three-space 2.3. Analysis of the solution . . . . . . . . . . . 2.4. The method of descent . . . . . . . . . . . 2.5. The inhomogencous wave equation . . . . . . . . 2.6. The Cauchy problem for general initial surfaces . . . . . . . . . . 2.7. Energy integrals and a priori estimates

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4 4 10 13 16 17 18

24

2.8. The general linear equation with the wave operator as principal part . 2.9. Mixed problems

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31

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35

3. Cauchy's Problem, Characteristic Surfaces, and Propagation of Discontinuities

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3.1. Notation . . . . . . . . . . . . . . . . 3.2. Relations between partial derivatives on a surface 3.3. Free surfaces. Characteristic matrix . . . . . . . 3.4. Cauchy's problem. The uniqueness theorem of Holmgren 3.5. Propagation of discontinuities . . . . . . . .

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4. Linear Hyperbolic Differential Equations . . . . . . . 4.1. Solution of the homogeneous equation with constant coefficients by Fourier transform . . . . . . . . . . 4.2. Extension to hyperbolic systems of homogeneous equations with constant coefficients . . . . . . . . . . . 4.3. Method of decomposition into plane waves . . . . . 4.4. A priori estimates . . . . . . . . . . . . . 4.5. The general linear strictly hyperbolic equation with constant principal part . . . . . . . . . . . . . . 4.6. First-order systems with constant principal part . . . . 4.7. Symmetric hyperbolic systems with variable coefficients . . xi

38 38 40 43 45 53 62

64 69 71

74 78 82 87

CONTENTS

xii

5. A Parabolic Equation: The Equation of Heat Conduction 5.1. Parabolic equations in general . . . . . . . 5.2. The heat equation. Maximum principle . . . . 5.3. Solution of the initial value problem . . . . . 5.4. Smoothness of solutions . . . . . . . . 5.5. The boundary initial value problem for a rectangle

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94 94 96 98

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101

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104

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6. Approximation of Solutions of Partial Differential Equations by the Method of Finite Differences . . . . . . . . . . 6.1. Solution of parabolic equations . . . . . . . . . 6.2. Stability of difference schemes for other types of equations Bibliography

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108 109 115 123

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131

Part II. Elliptic Equations, by LIPMAN BERS and MARTIN SCHECHTER

1. Elliptic Equations and Their Solutions

I.I. Introduction

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133

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131

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1.2. Linear elliptic equations . . . . . . . 1.3. Smoothness of solutions . . . 1.4. Unique continuation . . . . 1.5. Boundary conditions Appendix I. Elliptic versus Strongly Elliptic . Appendix II. "Weak Equals Strong" .

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134 135 139

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141

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143 144

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2. The Maximum Principle . . . . . . . . . . 2.1. Second-order equations . . . . . . . . . 2.2. Statement and proof of the maximum principle . . 2.3. Applications to the Dirichlet problem . . . . . 2.4. Applications to the generalized Neumann problem . 2.5. Solution of the Dirichlet problem by finite differences 2.6. Solution of the difference equation by iterations . . . . . . . 2.7. A maximum principle for gradients 2.8. Carleman's unique continuation theorem . . . . .

. . 3. Hilbert Space Approach. Periodic Solutions 3.1. Periodic solutions . . . . . . . . . . . . . . 3.2. The Hilbert spaces Hi . . . . . . 3.3. Structure of the spaces H. . . . . . . . . . 3.4. Basic inequalities 3.5. Differentiability theorem . . . . . . . . . . 3.6. Solution of the equation Lit =f . Appendix I. The Projection 'T'heorem . . . . Appendix II. The Fredholm-Riesz-Schauder Theory .

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150 150 150 152 154 155 158 160 162

164 164 165 167 170 174 175 177 183

CONTENTS

xiii

4. Hilbert Space Approach. Dirichlet Problem 4.1. Introduction . . . . . . 4.2. Interior regularity . . . . . .

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190 190 190 192 193

4.3. The spaces H' andH.' H. 4.4. Some lemmas in Ho . . . . 4.5. The generalized Dirichlet problem 4.6. Existence of weak solutions . . 4.7. Regularity at the boundary . . . 4.8. Inequalities in a half-cube Appendix. Analyticity of Solutions . .

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. 5. Potential Theoretical Approach . . . . 5.1. Fundamental solutions. Parametrix . . . . . . . 5.2. Some function spaces . . . . . . 5.3. Fundamental inequalities . . . 5.4. Local existence theorem . . . . . . . 5.5. Interior Schauder type estimates. . . . . 5.6. Estimates up to the boundary . . . . . 5.7. Applications to the Dirichlet problem . . . 5.8. Smoothness of strong solutions . . . . . Appendix I. Proofs of the Fundamental Inequalities Appendix 11. Proofs of the Interpolation Lemmas .

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196 198

200 202 207 211 211 216

220 228 231 235

237 240 242 250

6. Function Theoretical Approach . . . . . . . 6.1. Complex notation . . . . . . . . . . . . . 6.2. Beltrami equation . . . . . . . . . . . . . . 6.3. A representation theorem . . . . . . . . . 6.4. Consequences of the representation theorem . . . . . 6.5. Two boundary value problems . . . . . . . . . Appendix. Properties of the Beltrami Equation. Privaloff 's Theorem

254 255 257 259

7. Quasi-Linear Equations . . 7.1. Boundary value problems. 7.2. Methods of solution . . 7.3. Examples . . . . . Bibliography . . . .

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263 267

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282 282 284 286 291

Supplement I. Eigenvalue Expansions, by Lnxs GARDING

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301

Supplement II. Parabolic Equations, by A. N. MILGRA.tit

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327

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341

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Index

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PART I

Hyperbolic and Parabolic Equations By Fritz John

CHAPTER 1

Equations of Hyperbolic and Parabolic Types

A general precise classification of partial differential equations will not be attempted here. Instead some of the intuitive notions underlying the conventional definitions of Ipperbolic and parabolic

equations will be indicated. Roughly speaking, one wants to include in this class the differential equations of mathematical physics that describe time-dependent processes, and then also any other equations in which one of the independent variables, t, plays a role similar to the time. The solutions u of such equations can be thought of as evoliing with t from a given initial state under the influence of certain boundary conditions.

One imagines that the differential equation (or system of equations) describes the laws governing the changes of some physical quantities u defined in some portion R of x1, ... , x,,-space

for all values of t subsequent to an initial time to. In addition to the differential equations we are given certain initial conditions in R for t = to which constitute a compressed history of the system,

as far as it affects the future. There are also prescribed conditions

on the boundary S of R for t > to, which correspond to the influences on R of processes going on outside R, not already contained in the differential equations. Back effects of R on the outside are neglected. The problems customarily treated consist in determining a solution u of the differential equations which satisfies additional conditions of this kind. The boundary conditions usually state that certain expressions depending on u and some of its derivatives take on prescribed values on S for t > to; the initial conditions make a similar requirement in R for t = t,,. These prescribed values are denoted by functions f. Any problem in which the solution u of a system of differential I

9

FRITZ JOHN

equations is to be determined from data f is said to be well posed if u exists for arbitrary f, is determined uniquely, and depends continuously on f.1 Here the terms "arbitrary" and "continuous" have to be interpreted properly to make this definition of "wellposed" meaningful. For most purposes it is adequate to think of "arbitrary functions" as all functions of class C3 with some fixed

s,2 provided their values are formally consistent on the intersections of different surfaces on which they may be prescribed. Moreover, continuous dependence of u on f shall stand for the requirement that uniform convergence off and its derivatives of order < s shall imply uniform convergence of u. Physically, the data f in a well-posed problem constitute a sufficient and independent set of causes, resulting in a stable effect u. Mathematically, we have a continuous mapping from the space of data f onto the space of solutions u. In the case of hyperbolic and parabolic equations there exist such well-posed problems in which u for points x in R and -times

tl > to depends only on initial conditions in R for t = to and boundary conditions on S for to < t < t1. Evolution of u in time means that once u and its derivatives are known for t = tl we can find u for t = t2 > ti directly by considering the values at tl as initial values and using the boundary conditions for ti < t < t2, without further reference to the history of the system for to
ti must be given by (2.8)

u(x,t)

=.fi(x + c(t - tl)) , f(x - c(t - t1)) 2

1+ 2c z-ctt-c,)

gl() d

Since u(x,t) is determined uniquely this formula for u goes over

THE WAVE OPERATOR

7

into equation (2.6) when we substitute for f, and g, their expressions in terms off and g. Formula (2.6) shows that the value of u(x,t) depends only on

the initial values f(y), g(y) with y satisfying ly - xl < ct. The interval x - ct < y < x + ct constitutes the domain of dependence on the initial data for u at the point (x,t). Its endpoints are the intersections with the x-axis of the characteristic lines of slope l 1c and -1 /c through the point (x,t). Conversely the values off and g at a pointy influence u at the time t only at points x with Ix - yI < a. The domain of influence of the data at a point (y,0) is then the region in the upper half of the xt-plane bounded by the characteristic lines through the point (y,0). Iff and g have their support in a set S of the x-axis then u(x,t) for fixed t will have its support

in a ct-neighborhood of S.5 In this sense initial disturbances propagate with velocity c. In applications one is rarely concerned with pure initial value problems of the type (2.1, 2.5) in which the space variable x ranges over all values from - co to + oo. More commonly the coordinate

x which locates points of the physical medium is restricted to a finite interval 0 < x < L. Prescribed initial conditions (2.5) are then only meaningful for 0 < x < L. Additional information in the form of "boundary" conditions at x = 0, L for t > 0 is then needed to determine the solution. We treat first the simpler case of such a "mixed" problem in the semi-infinite interval 0 < x < oo. Let u(x,t) be a solution of class C2 of (2.1) for x > 0, t >- 0. We try to satisfy the boundaryinitial conditions u = f (x), for x > 0, t = 0 ut = g(x) (2.9) for x = 0, t > 0 u = h(t) Clearly necessary for existence of u is that f is in C2, g in Cl for

x z 0 and h in C2 for t > 0. Moreover if u is to be in C2 for x = t = 0 our data must satisfy the "compatibility" conditions (2.10) h(0) =f(0), h'(0) = g(0), h"(0) = c2f"(0) 6 The support of a function f is the closure of the set of points in which f 0. A function f defined in Euclidean space will be of compact support if it vanishes everywhere outside a bounded set. The ct-neighborhood of a set S consists of all points having a distance < ct from some point of S.

FRITZ JOHN

8

We shall prove that these conditions are sufficient to guarantee existence and uniqueness of a solution u. If u exists it is necessarily representable in the form (2.4) for x > 0, t > 0. The boundary-initial conditions (2.9) require the functions a, fi to satisfy the relations (2.11 a) a (x) + fl (x) = f (x), cx'(x) - cfl'(x) = g(x) for x > 0 (2.11b)

fort > 0

a(a) ; (3(-ct) = h(t)

If we normalize x and fi so that a (0) = X3(0) = z f (0)

we have uniquely

a (x) = 2f(x) '

2c

jg(t) d

Jg(E) d

9(x) = 2f (x)

,

for x > 0. Then by (2.11 b)

fi(t) = -a(-t) + h(-t/c) =

h(...t/c)

- f(-t)

fu

2

d5

for t < 0. Substituting into (2.4) shows that any solution u of our problem must be given by (2 . 12 a )

u(x, t )

=

f (x + ct) -I- f (x - Ct)

r

2

1

f x+Ct

2c Jx-rt g ($)

d

for x > ct (2.12b)

u(x , t) = h ( t

.r

+ f (x + Ct) - f (ct -

cl

x)

2 1

+ 2c-

z -i- Ct

et -x

dd

for x < ct

One verifies easily that this u actually is a solution of our problem, if f, g, h satisfy the regularity and compatibility assumptions made

before. In particular the compatibility conditions (2.10) are needed to make u and its first and second derivatives continuous along the "characteristic" x - ct = 0 (see Fig. 1).

Of special interest is the case where the boundary function

THE WAVE OPERATOR

9

h(t) vanishes identically for t >_ 0. A solution u(x,t) will then exist for x > 0, 1 > 0, if f (x) is in C2, g(x) in C', and the conditions

f (0) = g'(0) =f"(0) = 0

are satisfied. If we continue f and g for negative x as odd functions of x :

g(-x) = -g(x) f ( - x) = -f (x), formula (2.12b) becomes formally identical with (2.12a). The solution u(x,t) will be defined for all x and t >_ 0 by the common

Fig. 1

formula (2.6) and will itself be odd in x. We can consider u as extended by "reflection" on the l-axis. A similar procedure can be used to find the solution of the mixed problem for the finite x-interval; this consists in finding a solution of (2.1) for

t >0

0 -0

S2,u=0 It follows from (2.12b) that QrU(x>t) = 2 [

,t+rf (x) -

(x)] +

2c

5

Qz g(x)

d

for 0 < r < ct. Since by (2.17a) u(x,t) = lim 1 nru(x,t) r-.O r

we see that a solution u of (2.16) of class C2 with given initial values f, g is necessarily represented by the formula u(x't) (2.21)

a

ar

r,

+ u

1

c

dug

' f (x + ccy) dsd = 4-7r at t JJ IWi=

fJg(x

t l

I

1

-17

y) dsy

THE WAVE OPERATOR

13

2.3. Analysis of the Solution

We see that the solution u of the initial value problem (2.16, 2.20) is determined uniquely. On the other hand, for any f of class CA+3 and g of class Ck+2 formula (2.21) actually defines a function u(x,t) of class Ck}2 which satisfies the differential equation

and initial conditions, as is easily verified with the help of relation (2.18). The solution u depends continuously on the data in the sense that small changes in f, g and the first derivatives off will result in only small changes in u. The initial value problem is well-posed.

One feature of basic importance in the theory of general

hyperbolic equations in more than one space dimension has to

be emphasized. In contrast to the one-dimensional case the solution u does not depend continuously on the data f, g in the crude sense that small changes in f, g necessarily result in small changes in u. The occurrence of the differentiation sign preceding

f in formula (2.21) makes this plausible. An explicit example will show that the solution can actually behave worse than the data, or that initial disturbances can he amplified. It is verified easily that an expression of the form (2.21a)

=.(IxI -;

u(x,t)

Ct) +,3(1x1 - Ct) IxI

for arbitrary functions x(r), (3(r) of class C2 is a solution of the wave equation." These special solutions with spherical symmetry can be thought of as composed of progressing waves traveling towards or away from the origin with speed c, analogous to the situation in the one-dimensional case given by formula (2.4). Taking x(r) _ ,3(r) = 2r¢(r) where 0(r) is even in r, we find the solution O(ct + IxI) + ¢(ct -- IxI) -; ,

u(x,t)

=

ct

- 0(c1 -- IxI )

2 IxI

2

c(ct) + ctO'(ct)

¢(ct + IxI)

for x = 0

8 In general u is singular for x = 0.

forx0

FRITZ JOHN

14

which corresponds to the initial values

u =f(x) = 0(IxI),

at

= g(x) = 0

for

t=0

It is clear that small changes in 0 will result in small changes in the solution u for t bounded and x bounded away from 0, while the resulting changes could be arbitrarily large at x = 0, t > 0. If for example fi(r) converges uniformly towards the continuous function Ire _-11, the limit of the corresponding u would become infinite for x = 0, 1 = 1 /c. Physically, we have the situation where a singularity can arise at the origin from regular initial data through, focusing of spherical waves into the origin. It is plausible that such peculiar roughness of the solution will be restricted to focal lines or surfaces. On the average u should not be worse than initially. We shall see later that indeed the integral of the sum of the squares of the first derivatives,

the total energy, stays finite, if finite initially, so that singular behavior of first derivatives stays confined to sets of measure 0. The preceding analysis shows that for hyperbolic equations in

more than one space dimension no bounds for u at individual points can be obtained in terms of corresponding bounds for the initial values. If we only consider solutions u with initial f in C3 and g in C2 we are sure that a solution u in C2 exists; taking initial

data u and au/at at a later time ti, we find that they are only of classes C2 and C', respectively, and that construction of u(x,t), for

t > ti from data at tl becomes doubtful. If we want "persistent"

initial conditions, such that data at later times depend continuously on data at earlier times, changes being measured in the same way at both times, we have to make use of global quantities, like L2-norms, rather than magnitudes at individual points.

Formula (2.21) shows that the value of u at a point x at the time t depends only on the values off, g, and of the first derivatives

off all taken on the surface of the sphere of radius ct about the point x. This surface constitutes the "domain of dependence" of u on the initial data. Conversely, an initial disturbance f, g confined to a small neighborhood of a pointy will only affect the value of u at the time t for x near the surface of the sphere of radius ct about the point y. This illustrates the fact that for the wave equation L[u] = 0 disturbances spread with velocity c.

THE WAVE OPERATOR

15

The wave equation in three space dimensions has the additional atypical feature that the effect at a point x of an initial disturbance confined to a pointy disappears completely for ct > l y - xl, i.e., after the spherical wave from y has passed the point x. This

property, known as "Huygens' principle in the strong form" corresponds to the existence of sharp signals. It is absent for most

other hyperbolic differential equations; while a disturbance generally takes a finite time to reach a point, it usually persists after the wave front has passed the point.9 Even for the wave equation in three-space the strong form of Huygens' principle ceases to be valid if "obstacles" are present. Consider for example solutions of the wave equation defined in

the exterior of the unit sphere lxi = 1 which have vanishing normal derivatives on the sphere for all t. (If u is interpreted as a velocity potential this condition expresses the absence of flow

through the boundary.) Special solutions u of this type with spherical symmetry are represented by formula (2.21a) if the functions a(s) and fl(s) are connected by the relation fj(s) = a(2

- s) + 2

dE 2-R

If here a(s) has its support in a finite interval 1 < s s S the initial values of u will vanish for lx) > S. If Huygens' principle in the strong form were valid the solution u(x,t) would have to vanish at any given x with lxi L, 1 for all sufficiently large 1. Actually however we find for the value of u in a point of the boundary lxi = 1 u

1 f d

= 2a(l + ct) - 2e-1-c'

e`a($) dd t

For ct > -1 + S this reduces to u=

s

d

-2e-1-cc

1

! For a discussion of Huygens' principle see Hadamard [1], Baker and Copson [1]. For the question of characterizing equations for which Huygens' Principle in the strong form is valid, see in addition to Hadamard the papers by Stellmacher [1, 2]. For a more generally valid formulation of Huygens' Principle in terms of singular behavior on wave fronts see R. Courant and

P. D. Lax[1]andP.D.Lax[I].

FRITZ JOHN

16

While u decreases exponentially in time it does not have to vanish for large 00 2.4. The Method of Descent

The derivation of the solution of the wave equation in threespace given here was based on the use of a linear operator SZT with the property (2.19). Such operators can be constructed also for other dimensions, and can be used to solve the corresponding wave equations (or other equations with spherical symmetry)." To obtain the solution of the two-dimensional wave equation (a2

(2.22)

at2 -

C2

a2, - Cl a2,) u (xt,x2it) - 0 axj

ax.;

(equation of vibrating membranes) it is simpler to use Hadamard's "method of descent."12 Let it be the solution of (2.22) with initial values (2.23)

au

u =f (x1,x2),

at

= g(x1,x2)

for t = 0

Then u can be looked at as the solution of the initial value problem (2.16, 2.20) for the special case in which u, f, g are independent of x3. The solution is given by (2.21), where now

0r.f

r 4-r

Jjf(x1

H- 91, x2 + rY2) dSp,

I1I=1

r 2ir

f (x1

--I

ryl, x2

y1s_yyg 2r. By our preceding existence proof there exists a solution v(x,t) of

L[v] - N[v] = w(x,t)

THE WAVE OPERATOR

35

with vanishing initial data. Then iu - v is a solution of

v] = N[lu - v] with vanishing initial data. Moreover the constructed solution v has the proper domain of dependence and hence vanishes for

fix) >ct+2r Thus u - v has finite support in x, and hence vanishes identically.

Then u=vforlxi 0 which satisfies one wave equation in a region D of x-space and another wave equation with a different c outside of D. The values u and ut are prescribed for all x and t = 0. In addition, one has on the boundary B of D two transition conditions derived from conservation laws which state that au and d (du/dn) are continuous across B, where a and

i have different constant values in D and its complementary region.16

Explicit solution of such problems by quadratures are available only in a very few cases, e.g., where D is a half-space. For domains

with a more complicated geometry one usually falls back on the

method of separation of variables. One considers the special solutions u(x,t) which are simply harmonic in t with frequency co, i.e., the solutions of the form (2.63) u(x,t) = a(x) cos cot + b(x) sin cot = Re

[O(x)e-«']

Here 4(x) = a(x) + ib(x) is a solution, in general complexvalued, of the reduced wave equation (2.64)

0i + 220 = 0,

2 = co/c

Problems involving boundary conditions which do not depend on t are then formally reduced to problems for the elliptic equation (2.64). If, for example, u is a solution of L[u] = 0 for x in a domain D of the form (2.63) which is required to vanish on the boundary B of D, then the parameter 22 must be an eigenvalue A for D which can be taken and 0 a corresponding eigenfunction as real. The solution u for arbitrary initial values f, g is then represented formally by a series of the form x

u = 2; (a cos (c),t) ± b sin (cA t) c (x) n=1

(in analogy to (2.15)) where the constant coefficients a,,, b are determined from the initial conditions. Convergence of the series 16 See C. Muller [1].

THE WAVE OPERATOR

37

in the square mean follows from the completeness of the eigenfunctions." Other types of mixed problem arise in the study of scattering of waves by an obstacle. In the simplest situation one looks for a solution u of L[u] = 0 defined for x outside a bounded region D, which has vanishing normal derivative on the boundary D of B and whose initial conditions are those corresponding to a plane wave coming in from infinity in the direction of the x. ,-axis. Decomposing the incoming wave into components of the form e'l(zl-``) the problem is reduced to that of determining a complexvalued solution O(x) of (2.64) outside D with vanishing normal derivative on B, for which 0' = O(x) -- e1 ' satisfies the Sommerfeld radiation condition

lim

r(

r

- i A)

0

uniformly for

17 See Courant-Hilbert, Vol. I, Chapter VI.

1

CHAPTER 3

Cauchy's Problem, Characteristic Surfaces, and Propagation of Discontinuities

3.1. Notation

In this chapter we consider functions u(x) in an (n + 1)dimensional Euclidean space with points x = (x0, .

(3.1)

. . ,

For differentiation we use the operational symbol

Di=, a axi

(3.2)

i = 0,1,...,n

The D; can be combined into a gradient vector

D = (Do,

(3.3)

... ,

The general derivative of the function u(x) is of the form DooDil

(3.4)

with non-negative integers aa,

... D, ^u . .

... ,

. , a which can be combined

In order to keep formulae from becoming too cumbersome we make use of the notation introduced by L. Schwartz [1]. For two vectors into a vector a = (CEO,

S=

(S0, S1, . .

.

a = (ao, .

, Sn),

. .

, an)

where the a, are non-negative integers, we denote by

`

the

monomial (3.5)

Expression (3.4) is

boo

... fin"

then written simply as D°u. The order 38

CAUCHY'S PROBLEM

39

ao + al + ... + a of the derivative will be denoted by jai. In addition we define a ! by oc! al ! ...

In this simplified notation many of the formulae involving functions of several variables assume the same form as for functions

of a single variable. For example the power series expansion of an analytic function f (x) = f (xo, x1, ... , becomes

.f (x +y) = I

(3.6a)

i

yc'D°f(x)

For the special function f (x) = (xo + xl +

... + xn)k with a

positive integer k, formula (3.6a) yields for x = 0 k!

(3.6b)

Ixi=k 0C!

Leibnitz' rule for the derivative of a product of two functions takes the form (3.6c)

Dx(f (x)g(x))

_ Fr=

a!

Y!,

(DIf (x)) (D"g(x) )

The general linear differential operator of order m can be written in the form (3.7)

L=

A.(x)Dx IxI , (3.19)

... by

Lco> = L,

Lcx-ri>

=

(L(' )) cu = L(k) 0 - OLck>

21 The discussion is based on the treatment of the case of systems of first order equations given by Courant and P. D. Lax [1]. These authors also

study the case where u itself becomes discontinuous. For shock discontinuities propagating along noncharacteristic manifolds see Courant-Friedrichs [1]. 22 Sec John [5]. These formal derivatives of L also lead to useful forms of Green's identity.

CAUCHY'S PROBLEM

55

By induction Lck) is at most of order m - k. It follows that L(k) is identically zero for k > m and that L(") is a zero-order operator. By induction (3.20)

r!

r =O

r)

kk!

=I(-1)r

L(k)

)

(

and conversely k

Lok =

(3.21)

k

r!(k-r)! !

r

x rLcr)

One easily verifies that for two operators L1, L2 (L1L2)") = L(1)L2 + L1L(21)

More generally "Leibnitz' rule" holds: (L1L2)(k) =

(3.22)

k

!

ro r! (k - r) !

Lcr)k-r)

Let L1, L2 be respectively of orders ml, m2. It follows from (3.22)

for k = m1 + m2 that (LIL2)(m`+m2)

= mi + m2

t

' Llml)L2mq)

m1! mil!

Similarly for k operators L1, L2, .... Lk of orders m1, m2, (L1L2

...

Lk)cml+... +mk) _

(m1 + ... + mk) ! m1 ! m2 !

...mk

Lzml) L2 s)

!

..,mk

... Lk k)

If the factors Li are all of order 1 we find (L1L2

... Lk) (k) = k ! L(l) L.,l) ... L(')

Applied to a kth order derivative

D" =Do0...Dn^,

jai =ao +... +an =k

the formula yields

= k! (D(O'))"o ... (D;,'))" = k! rj'o ... 27n° = k! 2 This result permits us to find an explicit expression for the mth derived operator of the mth order operator (Da) (k)

77

L= IA1D" 1"I )D,,,_i)[i.C]

+ (m + 1) (D,)ci DiL(m)[AC] +

mm,D",_1)[AC]

+ (m + 1)77,L'111)[E]

FRITZ JOHN

58

Now for S characteristic there exists in addition to the column vector C satisfying (3.23) also a row vector I' for which

rA(x,i1(x)) = 0 and hence

on S rL("') = 0 Multiplying (3.24) from the left with q,r and summing over i (3.25)

we obtain (3.26)

)],rD,L('")[AC] ±

0=

m(1?l211'L(»,-1)[AC]

are first-order operators we have in (3.26) a linear partial differential equation of first order for the scalar A satisfied in the points of S. This equation is of the form

Since AD"') and

D"'-1)

(3.27)

i=u

a,D,[A] + bA = 0

with certain scalars a b. Here

Ii a,D, represents differentiation in a direction tangential to S. To prove this it is sufficient to show that 0

or equivalently that A _ 0 is a solution of (3.27) on S. Substituting for A in (3.26) we find indeed that ?j,FD,L(m'[0C] ± m

_

?j?rL(m-1)[0C] rL(m)[C]

n,rD=L`m'[c] + 1

1

cL(m-1)[C]) = 0

+m

Hence (3.27) represents a first-order linear differential equation for i. along the surface S. If we introduce the "bi-characteristic" curves on S defined by

(3.28)' = a Ids

CAUCHY'S PROBLEM

59

(i.e., the integral curves of the direction field on S given by the a,) we obtain an ordinary differential equation (3.29)

ds+bA=0

for A along each of those curves. The value of ). for one particular

s, i.e., at one point of a bi-characteristic curve uniquely determines A all along the curve. Thus (3.29) or (3.27) represents a law of propagation along bi-characteristics for the magnitudes of the discontinuity of the mth derivatives of u.

As an illustration consider again the case of the scalar wave equation (3.30)

L[u]

= (Do - c2(D + D2 + D2))[u] = 0

One verifies easily that here 3

(3.31) (3.32)

L(1>[u] = L[O]u

-1-

2 (Do/)(Dou) - c2

L(2)[u] = 2 (D00) 1 - c2

=1

(DO)(D1u)

3

t-1

(D.0)2)u

= 2(r12 - C2(rri + r2 + 7)a))u

Without restriction of generality we can assume that the surface S is represented by an equation of the form (3.33)

O(x) = xo - '(x1,x2,x3) = 0

Then

-

A()2)

(x ,y7) = L(2) = 2 ( 1 - c2 iG 1

The surface S is characteristic if L(2) = 0 on S, i.e., if V(x1,x2ix3) is a solution of the partial differential equation (3.34)

1 - c2 1=1

axl,12

=0

identically in x1, x2i x3. Since x0 does not enter explicitly in the expression for A(x,i1(x)) it follows that (3.34) implies that Lc2> = 0 for all x0, x1, x2, x3, not only on S.

In the present case of a single equation the vectors C and r

FRITZ JOHN

60

reduce to the scalar 1. The jump in any second derivative Di'u of a solution across S is then given by (3.35)

D#u'

- D'''u" = 27/''2

Since for characteristic S the expression L(2) vanishes identically, the differential equation (3.26) for A reduces to 0

=

Lci)[A]

=

2L[ o]

+

ai + c 21s

2

aXo

i

aa

1 ax; ax=

= di + 1L[0] ds

Here the bi-characteristic curves are defined by (3.36)

dd °

x,

= 2,

= 2c2 x

for i = 1, 2, 3

The bi-characteristic curves can be determined from a system of ordinary differential equations without reference to the particular characteristic surface S on which they lie. We shall not give the general proof of this fact. But in our example of the wave equation, we have from (3.34) along a bi-characteristic d 2c2 ds

aIP

ax;

=

3

dxk

a (2c2

k =o ds dxk `

= 4c4

a aX,

k1 a aak ax ,

=

2c4

R

(\2 _ ax i

k.1-0

aX k

/O

Consequently the dx,lds for i = 0, 1, 2, 3 are constant along a bi-characteristic curve. This means the bi-characteristic curves are straight lines in xoxlx2x3 space. Since by (3.34, 3.36) ' . (dX \2 c=1

ds

f2

= c2 t ` ds

they are straight lines forming the angle arc tan c with the x0-axis.

We can give a geometric interpretation in xlx2x3-space. Here

x0 is identified with the time t and S is replaced by its level surfaces s,

,(xl,x2ix3) = const. = t progressing with time. We think of a solution u with discontinuities in the second derivatives which are located in xlx2x3 space on the surface st, the "wave front." The characteristic differential

CAUCHY'S PROBLEM

61

equation (3.34) expresses that the wave front spreads with normal velocity c (see page 24). The bi-characteristics in four-dimensional space correspond to the rays in three-space given by

d' ^c2az

fori = 1,2,3

{

The rays are straight lines orthogonal to all surfaces s, The amount by which a second derivative of u changes at a point in three-space when the wave front passes is given by (3.35). Since the 71, are constant along a ray the jump in any second derivative is a constant multiple of A. Along a ray, the quantity A satisfies the ordinary differential equation

2d

+Ac2Aip=0

This formula can be interpreted geometrically as stating that the "intensity" A2 of the discontinuity varies inversely as the area of corresponding surface elements on the st cut out by the same rays.

We observe that the considerations given are purely formal and do not show that there actually exist solutions with discontinuities of the type described. It can be shown23 that the initial value problem for a hyperbolic equation with discontinuous

initial values leads to solutions that are discontinuous across characteristic surfaces which pass through the loci containing the discontinuities of the initial data. 23 See Courant-Hilbert, [1], Vol. 2, pp. 629 et seq.

CHAPTER 4

Linear Hyperbolic Differential Equations

We restrict ourselves to the case of a single scalar equation of order m of the form (4.1)

L[u] = I A(x)Du(x)

= w (x)

121:5M

Here x = (xo,xli ... , xn). The variable xo plays the role of the time t. Combining the remaining "space variables" x1, x2, ... , x into a vector X we write x = (xo,X). We make the assumption that the coefficients of the highest order terms in L are constant, i.e.,

for jai = m

A« = const.

Cauchy data shall be prescribed on the hyper-plane xo = 0: (4.2)

Dou(x) =fk(X)

forxo = 0,

and k = 0, 1,

..., m -

1

The initial plane shall be noncharacteristic, i.e., A(m.O....,o) : 0

Dividing by this constant we can normalize our equation in such a way that (4.3)

A("1,o,...0) = 1

In general no solution of the Cauchy problem will exist unless the coefficients are restricted by certain inequalities which express

the hyperbolic character of L. Solution of the problem in the hyperbolic case will follow the lines developed in Chapter 2 for the special case of the wave equation. 62

LINEAR HYPERBOLIC EQUATIONS

63

It is possible to reduce the problem for general data w, ft to the standard Cauchy problem

L[u] = 0

(4.4) (4.5)

(Dou),.=0 =0

fork =0,...,m -2:

(Do-1)s,=o =f(X) The reduction of the general problem to the standard problem is achieved by introducing first of all m-1 1

u* = u(x)

- 1k, fk(X)xo

as new unknown function. This function will have vanishing initial data and satisfy an equation of the form

L[u*] = w*(x)

(4.6)

with known w*. To obtain u * (x) we make use of Duhamel's integral zo

u*(x) =

(4.7)

V(xo,X,s)

ds

0

where V(xo,X,s) = v(x,s) is a solution of the differential equation (4.4) satisfying the initial conditions (DoV)x,=8 =

0

fork=0,1,...,m-2

w*(s,X)

for k = m

-1

which depends on the parameter s. In verifying that u* satisfies (4.4) and has vanishing initial data we make use of (4.3) and of ,the observation that Dxou*(x)

=foz°D0V(xo,X,s)

Dou*(x) =

X0

ds

fork < m

Do(xo,X,s) ds +

0

=J ='DoV(xo,X,s) ds + w*(xo,X) 0

Obviously the Cauchy problem for V can be reduced to a standard

FRITZ JOHN

64

Cauchy problem by a translation in the xo direction. The differential equation will not be changed by this translation in case none of the A"(x) depends on xo. We have in that case V(xo,X,s) = U(xo - s,X,s)

(4.7a)

where u(x) = U(xo,X,s) is a solution of the standard problem (4.4, 4.5) with f (X) = w*(s,X). Hence it is only necessary to consider the standard problem.

Even Cauchy problems for more general "spacelike" initial surfaces can be reduced to the standard problem by the methods used for the wave equation. It will be seen that the initial value problem for equation (4.4) can be solved by iteration once the corresponding problem for the "homogeneous" equation

Lm [u] = I A"D"u = 0

(4.8)

Ix1=m

has been solved and proper estimates obtained for its solution. This equation only involves the principal part

Lm = I A"D"

(4.9)

I"1- m

of the operator L, which, by assumption, has constant coefficients.

The standard Cauchy problem consists in finding a solution of (4.8) subject to the initial conditions (4.5).

u

4.1. Solution of the Homogeneous Equation with Constant Coefficients by Fourier Transform24

The customary approach to the solution of (4.8, 4.5) consists in finding special functions f for which the problem can be solved

explicitly, and then obtaining the solution for general f by superposition. Assume that we can solve the problem for all f (X) of the form f (X) = O(X, Y) depending on parameters Y = (yl, . . . , yn)

and let the corresponding solutions be u = v(x,Y); assume also that a "general" f can be written in the form (4.10) 24 See Garding [1].

f(X) = Jf(Y)0(X,Y) dY

LINEAR HYPERBOLIC EQUATIONS

65

with a suitable function f (Y). Then a formal solution of (4.8, 4.5) for f given by (4.10) is represented by (4.11)

u(x)

=f f (Y)v(x,Y) dY

This formula defines an actual solution of our problem, if the integral converges and if formal differentiation under the integral sign can be justified. Uniqueness of the solution follows by application of the method of Holmgren (see Chapter 3).

The classical method of Cauchy [1] employs Fourier transformation. This corresponds to choosing (4.12)

O(X,Y) =

n

(2i)-n12 eiF."1 ,

X. Y =

x=3'c

The function f (X) is then the Fourier transform off (X). For f sufficiently regular and vanishing sufficiently strongly at infinity we have the reciprocal formulae25 (4.13)

f (X) = (27r) _nh2 fe.t 2'f (Y) dY J

A

(4.14)

Y (Y) = (2i) -9:12 e- i T 1'f (X) dX

and the identity of Parseval between the Hilbert norms off and f: (4.14a)

fIfxI2dx = f If(Y)I2dY

For our purposes it is sufficient to observe that for f (X) in Co (i.e., for f (X) of compact support and writh continuous derivatives

of orders < s) the function f (Y) defined by (4.14) will be entire analytic in Y and will be such that 11,13f (y)

is bounded uniformly for all real Y. This follows by integration by parts. For s > n the integral in (4.13) will then converge absolutely and represent f (X). as See Courant-Hilbert [1], Vol. I, p. 80. When no limits are indicated integrals are to be extended over the whole space.

FRITZ JOHN

66

The first step is to obtain the solution u = v(x,Y) of (4.8, 4.5) corresponding to f = O(X,Y). We write

L. = I

(4.15)

P(D)

Ix) =:n

where P is a form of degree m in D = (Do, ... , D,,,), the characteristic form of the operator L. We try to obtain v of the type v(x,Y) = v(xo,X,Y) _ 0(X,Y)Z(xo,Y)

(4.16)

Using the special form (4.12) of 0 we find that Z(xo,Y) has to be a solution of the ordinary differential equation P(Do,iY)Z(xo,Y) = 0

(4.17)

with initial conditions (4.18)

(D Z(xo,Y))zo=o =

10

fork=0,l,...,m-2

t1

fork=m-1

The solution of this ordinary differential equation problem is given by

Z(x0,Y) =

(4.19)

1.

e'A°

2 Jr P iA,iY ( )

dA

where the integration is extended over a path r containing all roots A of the denominator of the integrand: P(iA,iY) = imP(A,Y) = 0

(4.20)

To verify that the Z of (4.19) satisfies the differential equation (4.17) we only have to observe that P(DO,iY) Z =

2t

dA = 0

since the integrand is regular analytic inside F. Moreover (

kZ):,=o =

1

Tff

r P(i),iY)

dA

has the values given by (4.18), as can be seen by expanding r to infinity and using that by (4.3) the coefficient of Am in P(iA,iY) is im.

LINEAR HYPERBOLIC EQUATIONS

67

A formal solution of (4.8, 4.5) is then represented by the expression

u(x) = u(xo,X) =

(4.21)

Z(xo,Y)f(Y) dY

(2,r)-"I2

This formula gives an actual solution, if the integral and the integrals obtained by differentiations of order 0. This means

that the H,[D"(u'Tk - u')] tend to zero for lal < m, ao < m

- 1.

FRITZ JOHN

80

-

Using Sobolev's lemma we can estimate D"(u'+k u') at any point in terms of H,[D"(u'+k u')], and hence also in terms of KS-l[u'+k - u']. It follows that for j--,. oo the D"u' converge uniformly for bounded x0 if lal < m, oco < m. The recursion formula (4.44) yields then convergence also for the remaining mth derivative D;.. u'. Thus the limit u of the u' exists, is of class C'" and represents a solution of (4.42, 4.43). Then u will also satisfy (4.1, 4.2) for IXI < r. We should like to have a solution of (4.1, 4.2) defined simul-

-

taneously for all X. It is sufficient to prove that the solution u(x) of (4.42, 4.43) that has been constructed does not depend on r for x restricted to a bounded set and r sufficiently large. Then

lim u(x) will be defined for all x with x0 > 0 and satisfy (4.1, 4.2).

The fact that u(x) is independent of r for large r follows from the existence of a bounded domain of dependence of the solution

of the Cauchy problem for the differential operator L. The Fourier representation (4.21) gives no indication that the solution

of (4.5, 4.8) only depends on the values of the data in a finite domain. We can however appeal to the uniqueness theorem of Holmgren since the operator L," has constant coefficients. Accord-

ing to that theorem a function u is determined uniquely in a region R covered by an analytic family of free hyper-surfaces SS with common boundary if we are given the values of Lm[u] in R and the Cauchy data for u on one of the surfaces S,. By assumption (4.3) the hyper-planes x0 = const. are free with respect to Lm. The same holds then for any surface if its normal incloses a

sufficiently small angle with the x0-axis. Thus there exists a positive constant c (only depending on the coefficients of Lm) such that a hyper-surface x0 = 0(x1,

. . .

,

¢(X) is free if

(0r)2 0 and let S2 denote the hyperboloidal surface

xo}a

-

a-IXYl2 1

',

Ace

X - Y 0 on the boundary D of a domain in X-space and also prescribe Au for

t =OinD.

94

THE HEAT EQUATION

95

ordinarily are of the form k

L[u]

= atk -

N[u]

with a dependent scalar or vector u = u(t,X) = u(t,x1, ... , X.)N is a differential operator containing no t-differentiations of order higher than k - 1.36 The prescribed initial data consist of the values of u and its 1-derivatives of orders < k - 1 taken at t = 0. All derivatives of u for t = 0 are then formally expressible in terms of the data from the differential equation. We deal with infinite propagation speed only if the order m of N is higher than k, i.e., when the plane t = 0 is characteristic. Otherwise the initial data would be the full set of Cauchy data and then, by Holmgren's theorem (at least for linear L with analytic coefficients), the solution u at a point is determined uniquely by the initial data in a finite region. Whether there actually exist well-posed

initial boundary value problems can be checked easily only in special cases where a priori estimates for the solution -can be given.

This is the case for certain second-order parabolic equations for which a maximum principle can be proved and for certain types of

equations with constant coefficients which can be solved by Fourier transformations'

A typical example of a second-order parabolic equation is furnished by the equation of heat conduction discussed below. More generally, equations of the type au

ar

= N[] u

have been considered where N[u] is an elliptic operator of second or higher order in the space variables. An example of a parabolic equation involving higher t-derivatives is furnished by the equation of transverse vibrations of thin plates 2

a2u

acs

+

22

a + axe) u = 0 1

37 See Rosenbloom [ 1 ], Gelfand and Silov [ 1 ], and the lecture by A. Milgram in this volume.

FRITZ JOHN

96

5.2. The Heat Equation. Maximum Principle

We consider the equation of heat conduction (5.1)

L[u]

_ `at - kA) u = 0

where k is a positive constant and u = u(t,xi, ... , xn) = u(t,X). An important property of this equation is the maximum principle which can be given the following form. Let R denote a closed and bounded region in X-space with boundary B, and let T be the cylindrical region in t, X-space described by

XER,

to 0 in the case where u is the absolute temperature), then u is determined uniquely by its initial values. 5.3. Solution of the Initial Value Problem

A formal solution of the initial value problem

L[u] = 0

(5.2)

for t > 0

u(0,X) =f(X) is obtained immediately from the Fourier integral formulae (5.3)

(4.13, 4.14) for f. Since e'x' Y-ktY-Y

is the solution with initial values e"', we find (5.4)

u(t,X) = (27r)-n12

f

f(Y) dY

= f K(t,Z - X) f (Z) dZ where (5.5)

K(t,Z

- X) = (2ir)-n f = (4irkt)

dY -n/2e-rs/4k

with r = IZ - XI. (The expression K represents the "fundamental solution" of the initial value problem (5.2, 5.3), with the initial value K(0,Z - X) = b(Z - X) where 6 is the Dirac function.) 39 See Tychonoff [ 1 ], Hormander [ 1 ], Hille and Phillips [ 11.

THE HEAT EQUATION

99

Instead of trying to justify the operations leading to the solution

f

- X) f (Z) dZ

u(t,X) = K(t,Z

(5.6)

we can verify directly that the u given by (5.6) satisfies (5.2, 5.3) by taking note of the following properties of K:

(a) K and its derivatives decrease exponentially for r - oo and fixed t > 0; (b) L[K] = 0 for t > 0 and any Z; (c) K(t,Z - X) > 0 for t > 0; (d) lim K(t,Z - X) = 0 uniformly for r z & > 0; t-.+o

fK(t,Z - K) dZ = 1 for t > 0 and all X;

(e)

lim f,>6 K(t,Z

(f)

- X) dZ = 0 for every d > 0.

Properties a-d are verified immediately. For the proof of property e we note that

JK(t,Z

- X) dZ = f K(t,Y) dY [(4rrkt)-112e-y,'/4kt] dy1 ... dyn =fJ ... IT i=1

IT[

" d ye]

=1

Property f follows from da,f u rn-1 f >a K(t,Z - X) dZ = fl,71=a r

dr

1

dcof fin, =1 -n/2 7T

f InI=1

rn-1(4nkt)-n/2e-rs/4td Go

an-1e-a' da

dco

-, 0

a/Vaxt

for

Let nowf(X) be continuous and uniformly bounded for all X. It is clear from properties a and b that (5.6) defines a u(t,X) for t > 0 which is in C°° and satisfies (5.2). Let M = 1.u.b. If (Z) 1. Given any e > 0 there exists a positive b = 6(e,X) such that

If(Z) -f(X)I ,4ktI

= (I + 02)''4ex"-.t"'4

+ 02),X' + OX" - Z)

where 0 =

t"ItI

It follows that

Iu(t' + it",X' + ix") < (1 + x

f If(Z)I K(t'(1 + 02),X' + OX" - Z) dZ

< M(1 +

02)n/4 ex X"/4k '

In particular for real positive t (5.8a)

f

Iu(t,X' + iX")I < ea" 14k: If (Z) I K(t,X' - Z) dZ

< 4je-''-

'14kt

If moreover f (X) > 0 for all X, then (5.8b)

I u(t,X'

iX") I
0.40 Let Sgt denote the mapping defined by (5.6) which associates with a function f (X), bounded and continuous in real X-space, the function u(t,X) depending on the parameter t. For real t > 0 the operator Sgt is highly smoothing; it converts bounded continuous functions of X into entire analytic functions of X which are bounded for real X.

For s > 0 we can consider u(s + t,X) as the solution of (5.1) with initial values u(s,X) for t = 0, since equation (5.1) does not contain t explicitly. It follows from the uniqueness of the initial

value problem for bounded u that the operators Sgt have the semigroup property

(5.9)

Sgt+8 = S242s

for s, t > 0

Since u(t + s,X) can be generated with the help of formula (5.6) either directly from the initial values u(0,X) or indirectly by way of u(s,X), we find by comparison the identity (5.10)

K(t ' s,X) = S)1K(s,X) = f K(t,Z - X)K(s,Z) dZ

which is the concrete counterpart of (5.9). The smoothing character of the operator Sgt makes it in general impossible to continue the solution of (5.2, 5.3) for negative t.

In order that u(t,X) could be continued backward in time as a uniformly bounded function the initial distribution f (X) would have to be analytic. A bounded continuous temperature distribution u given for t = 0 can always evolve in the future in accordance with the heat conduction equation (5.1) but cannot in general have originated from an earlier temperature distribution. The problem of determining the future is well posed, while that for the past is not. We can talk of "time's arrow" for equation (5.1).

The irreversibility of the operator Qt is connected with the fact that the differential equation (5.1) is not preserved if we '0 Analyticity of u follows from the uniform convergence of the integral (5.6) for bounded complex X, l with Re (t) bounded away from zero, and the analyticity of the integrand in X, t. As appears on page 107 a solution u for

fixed t > 0 is always analytic in X even without the assumption that u is bounded uniformly for all real X.

THE HEAT EQUATION

103

substitute -t for t. However with u(t,X) we have in v(t,X) = u(-t,iX) a formal solution of (5.1) with initial values v(O,X) = u(O,iX) =f (iX). This suggests the expression (5.11)

u(-t,X) = v(t,-iX) = fK(t,Z + iX) f (iZ) dZ ,1

= S2_tf(X) for extending u backwards in time provided f is analytic and such that the integral in (5.11) converges.41 It is easy to obtain necessary conditions which a function v(X) must satisfy in order that the operator SZ_t can be applied to it. Let v(X) be representable in the form v(X) = Sgt f (X) = u(t,X) where u is defined by (5.6) and f (X) is continuous and bounded. We have from (5.8a), for complex X, and real positive t (5.12)

Iu(t,X) I < Meh1"' (X )I E14kt

This inequality shows that v(X) is an entire function of X with a

certain growth restriction for complex arguments. Since, by Cauchy's formula, the derivative of a function in the center of a circle of radius r is at most the maximum of the function on the

circle divided by r, we find from (5.12) that, for real X, the gradient of u (i.e., the maximum of any directional derivative) is at most42

M ers1.ut

r Choosing the most favorable r, i.e., r2 = 2kt, we find that the gradient of a heat distribution v(X) which has existed for a time t and has been bounded by M is at most

41 See Widder [2], page 454. For numerical solution see John [I]. 42 A directional derivative of u(t,X) is given by (dl. u(t,X -{

34)/x=o

where 5 is a real vector with 1 = 1. We apply Cauchy's derivative estimate to u(t,X + ,34) as a function of the complex variable A. By (5.12) Ju(t,X + ).4)l < Me-r214

for 121 < r

FRITZ JOHN

104

Consequently an infinitely old heat distribution that has always been bounded by M, must be constant. We can obtain a still better result if we assume that f (X) z 0 for all X. Then from (5.8b), for complex X = X' + iX", eIIni ( 214ktu(t,Re (X)) = e1 '12I''s(X')

= I u(t,X) I < Applying the same arguments as before, we find that the gradient off (X) for real X is at most v(X) I

1 er2l4 t max v(Y) r 11'- XIsr

This inequality gives an estimate for the gradient of v(X) for positive u purely in terms of the age t of the distribution v and of

the maximum of v in a neighborhood of X, without a priori knowledge of the maximum M of u in the past. Thus observations of temperature gradient and temperature in the neighborhood of a point lead to an upper bound for the age t of a positive distribution if only heat conduction is involved.0 5.5. The Boundary Initial Value Problem for a Rectangle

So far we have dealt with the pure initial value problem for equation (5.1) in unbounded X-space. This is not quite a properly posed problem since, as was mentioned, there is no uniqueness

without some a priori restriction on the growth of the solution u(t,X) for large IXI. In this paragraph we shall solve the simplest initial boundary value problem, restricting ourselves to the case of a single space variable. Let u(s,z) be a solution of the equation 2

(5.13)

L[u] = as

- k aZ2 = 0

in the semi-infinite strip (5.14)

0 0. Since the first derivatives of v vanish at Ro and Mv(R0) > 0, at least one second derivative is positive, which is impossible at an interior maximum point. Hence, the maximum of v on Vo is achieved on So and thus at R (since v = u on So). Therefore the outward normal

derivative of v is non-negative at this point: 0 < (du/dN) + e(dg/dN). Thus du/dN > 0. The desired function g can be written down explicitly: g = e_2 - e-2x02

where a > 0 is a constant, r is the distance from the center Rl of the two spheres, and ro is the radius of the outer sphere. We

L. BERS AND M. SCHECHTER

152

have (assuming xi to be the origin and denoting the radius of Si by r,) Mg = e 2r2(4x21 a,,x;x, - 2aj a,x{ - 2al a,1) > e-"r2(4xzmr, - 2nxKro - 2nxK) so that Mg > 0 in ,, if x is sufficiently large.

Now we can prove I. Assume that the maximum ,u of u is achieved at an interior point x0 and let 6 > 0 be so small that the ball Ix - x01 < 26 lies in U. We claim that u(x) = u for Ix - x I < 6. In fact, assume that u(xl) < u, Ix1 - x01 < 6.

Then there exists a number p, 0 < p < 6 such that u < µ in the ball Ixl - xl < p and u = u at some boundary point X of this ball (IX - x1 I = p). By the part of III already proved some directional derivative of u is positive at z. This is impossible, since

all first derivatives vanish at an interior maximum point. Now let A c

0. By I, u = u(x0) in a neighborhood of x9. Thus the set of points at which u = u(xo) is open. Since it is closed by continuity, it must coincide with the whole domain 9.

For the proof of IV, assume that k is a point on I with = max u = u(.x) > 0. Then Mu = Lu - au > 0 near z and, by III, either du/d1V > 0 at X or u - µ at some interior points close to X. In the latter case u is constant by II. 2.3.

Applications to the Dirichlet Problem

We shall now discuss some applications of the maximum principle to boundary value problems. Consider first the Dirichlet problem u = O on (3) Lu =f in Vt, where f, 0 are given bounded functions.

THE MAXIMUM PRINCIPLE THEOREM.

153

If a < 0 then the solution u of the Dirichlet problem (3)

satisfies the inequality

Jul < max 101 + (e" - 1) max If I

(4)

where a = (11m) [K -{- (K2 + 4m)112] and d is the diameter of 9.

The interesting thing about this inequality is that it does not depend upon the size of a, or on the continuity properties of the coefficients, or on the shape of the domain.

An immediate corollary of this estimate is the uniqueness theorem for the Dirichlet problem. If f = 0 = 0, a < 0 and (3) holds, then u = 0. Of course, this also follows directly from II. In order to prove the estimate we may assume, without loss of

-

generality, that 9 lies in the strip 0 < x1 < d. Set g(x) = e' eu'. In the strip considered, we have that e"d - 1 > g > 0 and Lg = (a11a2- a1x)e'zl + ag < -mat ; Kx = -1. Now set

-

h =maxI0I +g(x)maxIfI.Then Lh =maxIf(Lg+amaxI0l -max If I and h > max 101. Hence v = u - h < 0 on 9 and Lv=f - Lh >f -}- maxIfI >0 in W. By II, v :!5: 0 in 1.

Similarly, u ! -h. Hence Jul < max 101 - max If I g(x) which implies the assertion. The proof shows that the theorem can be considerably strengthened.

If a is not nonpositive, we cannot expect in general that the Dirichlet problem will have a unique solution. Nevertheless, the

uniqueness theorem can be proved and an estimate can be obtained if the domain is sufficiently small. THEOREM. Assume that a < k where k is a positive number, and that the diameter d of 0. By II, z

must be a boundary point. Near this point Mu = -au > 0 so that the normal derivative of u must be positive at f by III. But

all tangential derivatives vanish at X. This implies that the prescribed derivative au/aa = 0 at x, which is impossible.

THE MAXIMUM PRINCIPLE

155

2.5. Solution of the Dirichlet Problem by Finite Differences

As the next application of the maximum principle, we shall discuss the effective numerical computation of a solution of the Dirichlet problem (3) by the method of finite differences. For the sake of simplicity, we restrict ourselves to two dimensions and to a differential equation of the form

Lu = Du + alu, + a2u,, + au =f

(6)

The functions al, a2 and a are assumed to satisfy the conditions

aS0 We approximate the differential operator L by the finite $a11 + 1a21 < K,

difference operator

Lhu=Mhu+au

(7)

= h-2[u(x + h, y) + u(x - h,y) -{- u(x,y + h) + u(x, y - h) - 4u(x,y)] u(x + h,y) u(x h,y)

-

+ a1(xy)

2h

u x, y

u x, y + a2 (xy)

-

+ a (xy) u (x,y)

2h

the mesh width h being a small positive number. Note that Lhu

Lu as h

0 for every C2 function u.

In order to formulate a boundary value problem for this difference equation, we introduce the following notations. The points

(x + h, y), (x, y + h), (x - h, y), (x, y - h) will be called the h-neighbors of PO = (x, y) and will be denoted by P01, P021 P03, P. Py situated in 9, A lattice domain'h is a set of points Pi, P2,

....

having coordinates which are integral multiples of h, and such that for every point in the set its h-neighbors belong to K We also require that given any two points Qo and Q00 in 'h there should exist points Qo = Q1, Q2, ... , Q, = Q00 in Wh with Qj

an h-neighbor of Q,_1. Neighbors of points of Fh which are not

themselves points of Th form the boundaryh of !Yh; these points will be denoted by PV+1, PV+21 ... , P' ,-sr. The union of

L. BERS AND M. SCHECHTER

156

yh

and 'h is the "closure" 1h of !Ye,,. We assume that h is so

small that a lattice domain 65h exists and that

hK < 0 < 1

(8)

For the difference equation we now pose the following Dirichlet problem (9)

Lhu

=fatP1,P2,...,P.V;

u =0 atP.,+1,.

.

, PV+.11

The solution u is of course to be defined only at the N -i- M points P,.... , Pv h,,, or as we say, on the lattice 7h.

We claim that the following maximum principle holds for functions u defined on the lattice domain h. If Mu > 0 in 1,

then either u is a constant or it achieves its maximum on the boundary The proof is trivial. For a point Pi in F,, the condition 0 reads Mhu

u(Pi)
0 in,, cannot have an interior positive maximum unless it is a constant. Furthermore, the same reasoning which led to the estimate (4) leads now to the following estimate for the solution of the Dirichlet problem (9) : (10)

Jul < max 101 ; C max If I

THE MAXIMUM PRINCIPLE

157

where C depends only on K, the domain T, and 0. In fact, we may use the same comparison function as before since we have that Lh(ead - exz) _ -a2 e,,x(sinh (ah/2))2 ah/2

-blot e"x sinh (ah) + a(e"d - e"s) /2 )12l(1

(sin

- ot2 a

ah/2

J

- K ah coth (ah/2) J < - 1

if a > 0 is large enough, provided that 9 lies in the strip 0
0. It suffices to show that w(z) = 0 for Izl < R, R > 0 and small Proof.

THE MAXIMUM PRINCIPLE

163

enough. The functions w(z) n = 1 , 2, ... , vanish at z = 0 and satisfy (16). Applying (17) to the domain 0 < I zI < R, we see that max I z-"w(z) I< kR-" max I w(z) I Izj 0, Ht is the space of periodic functions having

generalized L2 derivatives up to the order t.

In particular, a periodic function of class Ct belongs to H. A partial converse of this statement is also true. LEMMA 3.

(Sobolev [1]). If u(x) E H, and t > [n/2] + k + 1,

then u is of class Ck, and (17)

max IDDul < const. Ilullt

for IpI < k

L. BERS AND M. SCHECHTER

168

It will suffice to prove this for k = 0 (cf. (10)). Now if u = E afe'f r c Ht, t > [n/2] + 1, then

Ixfl)I0

these inequalities, combined with (23), yield

(u, Lou + Liu) > ci

11'

t

- cl

(ci > 0)

1 1u1 1u

Case (c). L = Lo + Li + L2 where Lo and Ll are as in Case (b)

and L2 =

aD(x)DP. By integration by parts and by the Ipl 0 For small a this together with (24) yields Garding's inequality. General Case. For a sufficiently small n > 0 we construct e const. Ilu

(e-"'/2) const. Ilu 112,

periodic C. functions aq(x), (u2(x), . . properties. (i) On the support of

.

,

(o).\-(x) having the following

each w, the oscillation of IpI = m, of L is less than 71.

each leading coefficient (ii) E (i)1(X)2 - 1. By Case (c) (26)

((u,u, Lo);u) o > pos. const. II oo;u Il ,,,2 - const.

II (,),u II 2

but (u,Lu) 0

(27)

f

= (I (1 )uLu dx =

((o,u, L(o;u)o ± R

R = -1 f(f))u {L(oju - (u,Lu) dx Integration by parts yields

R=J

dx,

IpI + 191 < m,

with some fixed C,, functions

141 < m/2

and as before (e-m/2)

IRI < e const. Ilu IIn,/l +

(28)

I pI < m/2,

const. Ilu Il0

Clearly II(,),u Ilo pos. const.

II U

II71t/2 - const. Ilu 112

Combining relations (26-30) we get Garding's inequality. The use of Fourier series (or Fourier transforms) seems indispensable for the proof of Theorem 1. But in the classical case m = 2 integration by parts suffices. Indeed, assume that

L-L1 L1

= -5 alk(x)

a2

a.r,ax; '

,

L2 a

L2 = Y a,(x) ax, - a(x)

HILBERT SPACE APPROACH, I

with E aik(x)

173

c E 6rj, c > 0. Then

au au

= SJ a,k ax. axk dx +

(u)Liu)o

a

2

>cI

If ax

au ku

t

dx

x

-con st. Ilullo Ilulll

11TIUxillo

pos. const.

1 1 U1 1i

- const.

11U 11

o

and since I (u,L2u) I is easily seen not to exceed e const. IIu II

(1/e) const. IIu Ilo for every e > 0, Garding's inequality follows.

-

At the moment we shall prove Theorem 2 only in a weakened form. More precisely, we shall show that (22) holds for A > A0(t), A0(t) depending on t and L.

Let s be a fixed non-negative integer. The differential operators K"L and LK" of order m + 2s are elliptic; hence, by Theorem 1, there are positive constants c;, c., depending on s such that Proof.

(u,K"Lu)o

:2! c'l Ilu ll8+,,,/2

(u)LKsu)o ? c1

IIu

112

- C2 Ilullo - c2' IIu Ilo

Using the first inequality together with (15, 6, 14, 4, 3) we have that flu II 8 I m/2

II Lu + Au II 8-m/2 =

IIuII "+m/2

II K3Lu + AK"u ll -s-m/2

> (u, K"Lu + ).K"u)o >- ci IIUII8+m/2 - c2 Ilullo + A(u,Ksu)o

=

ci flu Iii+m/2

- cz IIu Iln + )'11U112 ? ci

IIu IIS+m/2

+ (2 - c2) Ilullo ? ci Il u ll9 i m/2 for A > c2 and dividing by 11U 113. m/2 we obtain (22) for t = s + m/2. Similarly, IIuII-s-m/2

IILu + Aull_s-m/2 = IIK-'ulls-,,,/2 IILu + Aull-s-m/2

> (K-"u, Lu + Au) o = (K-"u, LK"K-"u + Au) IIK_"u112

ci

= ci

11K-"ull8+m/2 - c2

IIUI12

8+m/2 - c2

IIUII

+ A IIuI12-8

-2" + . IIuII? ? ci Ilullo 8+m/2

for A > c? and dividing by IIu II -s 1 m/2 we obtain (22) for t = -s + m/2.

A complete proof of Theorem 2 will be given later.

L. BERS AND M. SCHECHTER

174

3.5.

Differentiability Theorem

Theorem 2 (in its weakened form) implies LEMMA 11.

For every t and for A > 0 sufficiently large the operator

L + A is a bounded linear one-to-one mapping of H, onto the whole of H,-,n and the inverse mapping (L + 2) -1 of H,_m onto Ht is also bounded, independently of A.

Proof.

By Lemma 9 the linear mapping L + A: Ht -* H,-,. is

bounded. If 2 is so large that (22) holds (at the moment this choice may depend on t), then Lu + Au = 0 implies u = 0, so that the mapping is one-to-one. The inverse mapping is defined on

the range R, = (L + A) (H,) and is bounded by virtue of (22), the bound being independent of A for A > A0(t). Next, R, is closed. Indeed, assume that vi E R, and II v, - v 0. Then v! = Lu, + Au,, u, e H. By (22) we have that II u, - uk II t S c3(t) II Vi - Vk II t-m , 0 for j, k ---+ oo. Hence there is a u-E Ht such

that Ilu, - u II, -> 0. By Lemma 9, IILu1 + Au, - (Lu + Au) II t-m

0. Hence Lu + Au = v and v e R. In order to prove that R, = H,_,n we use the adjoint operator L* (cf. Chapter 1). We note that (31)

(u,Lv)o

- (L*u,v)o = 0

for periodic C,, functions (proof by integration by parts) and hence

(by an obvious limiting argument) also for u c- Hs, v e Ht with

s+t>m.

Assume that A is so large that (22) holds also for L* and that R, 0 H,_r. Then there is a w e H,_,n such that w 0 0, (w, Lit + Au),-,. = 0 for all u in Ht (projection theorem, cf. Appendix I to this chapter). Hence 0 = (Kt-mw, Lu + Au)o = (L*Kt-mw + AKt-mw, u)o for all u in H,, and in particular for all C,,, functions u. This implies

(for instance by (31)) that L*Kt-mw + AKt-mw = 0. Now, Kt-mw e Hm_, and if A is large enough, (22) applied to L* shows

that K'-mw = 0. Then w = 0, contrary to our assumption. THEOREM 3. If u is a periodic distribution and Lu a H u EHs+m

then

HILBERT SPACE APPROACH, I

175

This is a differentiability theorem. It implies that every (periodic) solution of Lu =f is a function if f E H_,,,, has L2 derivatives up to the order k if f e Hk_,,,, is of class C, if f e Hs with s >_ [n/2] + r + 1 - m (Lemma 3), and is of class C,,,, iff is. In the next chapter we shall extend this result to nonperiodic solutions.

Set Lu ==f. By Lemma 7 the distribution u belongs to Hk for some k. Hence f + Au E Hm1n (k,8) and if A is large enough u = (L + A)-' (f + Au) belongs to Hmin (k+m,s+m) Proof of Theorem 3.

Repeating the argument we see that u c- Hmin (k+im,s+m) for j = 1, 2,.... Thus u c- H,,+).,,. 3.6.

Solution of the Equation Lu = f

We shall show now that the equations (32)

Lu =f

(33)

L*v =f

form a Fredholm pair.° Here f is a given periodic distribution;

we already know that for f EH, every (periodic) solution belongs to He. In particular, all solutions of the homogeneous equations

Lu = 0 L*v = 0

(34)

(35)

are Cr,, functions.

Let A0 > 0 be so large that the bounded linear mappings M = (L + AO) -1: Ho H,,, and M* = (L* + An)-1: Ho --* H. exist (Lemma 11). Since H,,, c Ho we may, and shall, consider

M and M* as mappings of Ho into itself. By Lemma 10 these mappings are completely continuous. The homogeneous equations (34, 35) may be written in the form

u-k0Mu=0, 6 See Section 1.5.

u-A0M*u=0

L. BERS AND M. SCHECHTER

176

Furthermore, the operators M and M* are conjugate in Ho. Indeed, by (31), we have that (Mu,v)o = (Mu, L*M*v + AOM*v)o

= (LMu + ).0Mu, M*v)o = (u,M*v)o

Hence the Fredholm-Riesz-Schauder theory is applicable (cf. Appendix II to this chapter). We obtain THEOREM 4.

The homogeneous equations (34, 35) have the same

finite number of linearly independent solutions. THEOREM 5.

The equation Lu + Au = 0 has nontrivial solutions

only for a denumerable set of values ) with no finite accumulation point.

Indeed, the equation considered may be written in the form

u-A'Mu=0,with 2'=io-2. THEOREM 6.

Equation (32) is solvable if and only if (f,v) o = 0

for every solution v of (35).

Assume first that f e H, for some s -,,>- 0. Then equation (32) is equivalent to the following equation in Ho: Proof.

u - AOMu = Mf By the general theory, it is solvable if and only if (Mf,v)o = 0 whenever v - AoM*v = 0. But this is the same as saying that (f,v)o = A0(f,M*v)o = 2o(Mf,v)o = 0 whenever v - ,10M*v = 0. Since v - A0M*v = 0 is equivalent to L*v = 0, the result follows in this case.

Assume now that f e H , for s > 0. Let t be an integer such

that m + 21 > s. Since LK' and K'L* are adjoint elliptic operators of order m + 2t, there is a Al such that LK' + ,11 and K'L + Al are one-to-one linear mappings of Hm±l,_, onto H ,. It is easily Set M' _ (LK' + 21) -1, M'* = (K'L* + shown, as in the discussion preceding Theorem 4, that M` and M'* are conjugate in Ho. We now note that (32) is solvable if and only if (36)

LK'w = f

has a solution. Now (36) is equivalent to (37)

w - a1M'w = M !f

HILBERT SPACE APPROACH, I

177

and My E Hm+2t-s c Ho. Hence we may again apply the abstract theory to conclude that (37) has a solution if (M f v)o = 0 for all v E Ho such that v - AIM(*v = 0. This is the same as saying that (32) has a solution if (f,v)o = A1(f,Mt*v)o = A1(M f,v) = 0 for all solutions v of (35). Thus the theorem is proved in this case as well.

Combining Theorems 4 and 6 (or reasoning directly from the general theory) we get Equation (32) is solvable for every f c- H ,, if and only if (34) has only the trivial solution u = 0. THEOREM 7.

We can now complete the proof of Theorem 2. We first note that from Garding's inequality (21) it follows that for some A > 0, Lu + Au = 0 implies u = 0 whenever A > A [e.g., take A = c2 in (21) ].' Now assume that Theorem 2 is false. Then there exists, for some fixed t, a sequence {u5} in Ht and a sequence of numbers A; > A such that II u; IIt = 1, IILu, + A,uf llt_m , 0. Let Ao be so

large that the bounded mapping (L + A)-1: Ht_.,,, , Ht exists for 2 >_ Ao (Lemma 11). Clearly A, < A. and we may assume, A >_ A. By Lemma 10 selecting a subsequence if need be, that A, we may also assume that {u,} is a Cauchy sequence in Ht_,,,. Now

ui = (L + Ao)-1(Lu, -- A,u,) + (A0 - 2,)(L + Ao)-lu1 so that 0 for j, k -- oo. Hence there is a u in Ht with II us - Uk Il t 0 and we have : Il u ll = 1, Lu + Au = 0 which is 11u1 - u II t impossible.

The same method shows that whenever A is not an eigenvalue

(i.e., whenever Lu + Au = 0 implies that u = 0) (L + A) -': Ht is a bounded operator, the bound being uniform Ht_m on every closed set of A's not containing eigenvalues. Appendix I. The Projection Theorem

We shall consider some simple theorems in a real Hilbert space H. (H consists of elements u,v, ... , for which the operations of addition and multiplication by real numbers are defined and these 7 Recall that Lu + ..u = 0 implies, by Theorem 3, that u is a C,, function. Hence the GArding inequality can be applied.

L. BERS AND M. SCHECHTER

178

operations obey the usual rules of a vector space. Moreover, for every pair of elements u,v there is defined a real number (u,v) called the scalar product in such a way that (u,v) = (v,u), (u + v, w) =

(u,w) + (v,w), (Au,v) = A(u,v), (u,u) > 0 for u ; 0 for all u, v, w c- H and real number A. In such a case we can define a norm 11u II = 1/(u,u), and it is easily shown that (u)v)I S IIuII IIuII 11u + v11 0 as m, n -- co, there is a u E H such that 11 u,, -u11 -*0asn- oo.) A subset S of H is called a subspace if Au + ,uv is in S for all real numbers A, a whenever u and v are in S. It is closed if ueHand Ifu, -u11 ->0asn- imply that u E S. First we prove

c S,

Let M be a closed linear subspace of H. Then for every u c- H not in M there is a v e M such that LEMMA 1.

IIu - v 11 = g.1.b. 11u - w II WEM

Let d = g.l.b. flu - w 1j, w c- M. Then there is a oo. From (1) sequence c M such that IIu - w, II , d as n Proof.

we see that 4 Ilu - (wm + wn) III + IIWm - wnII2

=

2( IIu

- w. 112 + Ilu -wn 112) -' 4d2

as m,n -> oo. Since-(w,. + w,,) c- W, 4 IIu

-

(wm + wn) II2 > 4d2

- wn II -' 0 as m, n -> or,. Since H is complete, there is a v c- M such that 11 wf, - v 11 - 0. This means that and hence

11 wm

IIu - v 11 = lim flu - wn 11 = d.

HILBERT SPACE APPROACH, I

179

(Projection Theorem). Let M be a closed linear subspace of H. Then for every u e H, u = v + w, where v e M and (w,M) = 0 [i.e., (w,h) = O for all h e M]. THEOREM 1

Proof.

If u c- M, set v = u and w = 0. If u is not in Al, then

by Lemma 1 there is a v e Msuch that Ilu - v II = d, the "distance" from u to M. Now if f is any element of M

IIu-vll2-22(u-v,f)+22Ilfii2

IIu - L'IIZC llu - U - ?/ II2= for all real 2, in particular, for

2=(u-v,f) Ilf112

Thus

Ilu - vl12 < Ilu - vll2 -

2(u

IIf112

f)2 +

(u

f

II,f)2

and hence (u - v, f )2 < 0, which means that (u - v, f) = 0. Since f was any element of M, w = u - v meets the requirements of the conclusion of the theorem. COROLLARY 1.

If M 0 H, there is a nonzero element w of H such

that (w,M) = 0. Proof.

Take u not in M. Then by Theorem 1, u = v + w,

where v e M and (w,M) = 0. Clearly w

0.

A bounded linear functional Fu on H is a real valued function on H

which satisfies the following conditions: F(ui + u2) = Ful + Fu2; F(2ul) = 2Fui, 2 a real number; IF(u) I < K IIu II for all u e H and some K > 0. The smallest K for which the last statement holds is denoted by IIFII

THEOREM 2 (Representation Theorem; Frechet, Riesz). For every bounded linear functional Fu on H there is a uniquely determined element f e H such that Fu = (u, f) for all u e H and IIFII = IIf1I Proof.

Let N be the set of all v e H such that Fv = 0. N is a

linear subspace of H. For if vl and v2 are in N and Al and 22 are any real numbers, F(2ivl + 22v2) = ).1Fvi + 22Fy2 = 0. Moreover, Nis closed in H. For if {v,,} is a sequence in N and II v,, - v II -0, theniFvl and V EN.

180

L. BERS AND M. SCHECHTER

Now if N = H, the theorem is easily proved by setting f = 0. Otherwise there is a w 0: 0 in H such that (w,N) = 0 (Corollary 1). Therefore Fw - 0 and for any u E H

FIu

-Fwwl =Fu -FwFw =0

and hence u - Fw w is in N. This means that

Iu -F-tww,wJ

=

0

i.e., that (u, ') =Iw 11w112

Therefore

Fu=

(u

w Fw)

' IIwII2

1

and (w/III' 112)Fw is the element f in the conclusion of the theorem. If f' also represents F, then Il f _f'112 = (f - f ', f -fl) _

(f -f',f) - (f - f',.f ') = F(f -f') - F(f -f') = 0. Next, Ilf III = (,f f) = F(f) IIFII Hence IIFII - E < (g,f) < Il f ll. Thus IIFII = 11f 11.

- E.

The following theorems will be found useful later. THEOREM 3 (Lax-Milgram [1]). Let [u,v]

real valued function defined for pairs of elements in H which is linear in both u and v. Suppose it satisfies (2)

I[u,v]I < lull

(3)

Ilu 112

be a

IIvII

< K[u,u]

for all u, v in H. Then for every bounded linear functional Fu on H there is an f c H such that

Fu = [u,f] For fixed v e H, we have, by (2), 1 [u,v] I < const. II u II Hence [u,v] is a bounded linear functional in H. Thus (by Theorem 2) there is an element Sv in H such that Proof.

(4)

[u,v] = (u,Sv)

HILBERT SPACE APPROACH, I

181

Obviously, Sv is a linear mapping of H into itself. We observe that by (2) IISuII 2 = (Su,Su) = [Su,u] S IISuII

.

Hull

so that 11sull

0, then

LEMMA 2.

Ilull

[n/2] + k + 1, then u is of class

and max I Dpu l < cont. II u Il t Ipl 0 and set X" > 0

v1(x',xn) = v(x',xn)

_

8-}-1

x,, < 0

.Z.v(x', k=1

... , xi_,) and s = max (m, t ± 1). The constants

where x' = (x,,

Ak are so chosen that v1 is in C. in 2, i.e., that RTL

(21)

k=1

j = 0, 1, ... , S

(-k)'2,, = 1

Thus v1 has compact support in .2, and ID'vlz dx

ID11v1I2 dx
m 1), it follows from (22) that (20) holds for v.

-

It remains to prove that for t < m - 1

,, j h > 0. Letting a -o 0, we have Set s = max (m, t , 1) and suppose that v is in C, n HO in E and vanishes near all faces except x = 0. Assume that D,v E Ht in for i = 1, ... , n - 1, and that IILvll,,,-t+1 is finite. COROLLARY 3.

Then v E H`+1 in 2.' and (26) holds.

We use induction on I and show first that the assertion is true for t = m'. Let a' denote any difference quotient in the x, direction, i , n. Then Proof of Theorem 9.

L,,,r

L8;'v

where L,,, is the operator whose coefficients are the difference quotients of those of L. Thus IILb;'z'il

,n

IIa (Lz')'1 0 (cf. the end of

Appendix II to Chapter 1). Hence, by Lemma 7, D,v E Hm' in S2,_ and IID,vllm' s C'

Letting E -- 0, we see that D,v E H7" in Q,,. Once this is known, we

can apply Corollary 3. We therefore conclude that v e Hm'+i in QT.

Now assume that the theorem is true for t - 1 > m'. We have

LD,v = D,Lv + L'v where L' is an operator of order r + h. Now by Sobolev's inequality (cf. Lemma 3 of this chapter) there is

an integer N (>m) such that max IvI < x(n,N,R)

(4)

Izi _x

Employing the fact that Ivl g,1; and IIVII:,R are equivalent norms

for test functions (Lemma 5 of this chapter), we see that (3) implies (5)

C Const. (ILvi \.-m,r

IIVII.\',r+h

!h

+

Ivl.V-.,,+h)

Now if Isl = N - m, there are C,, functions bxo such that

DBLv = D3Lw + Hence (6)

.1 p'to

bD,D.Dv'4DD"w

N

II wll v,r-h
2) Oc2 log IXI = b(x) (n = 2) where b(x) is the symbolic Dirac function, i.e., a distribution defined by the relation Ac.,

Ix12-n

= b(x)

J(x)6(x) dx = 0(0) for every test function O(x). Now (la) follows immediately from (1), since O(x - z) dz = f ao(x - z) o r 0(y) dy = ASJ dz x

IX

IZI1--2

-yln-2

=f AM ') IX

yj 11

2

dy

J

IZIn-2

= c 'fi(x)

From (1) we obtain, integrating by parts, the identity (2)

O(x) = const.

y' a4 dy

f X` _ Ix J -yI" ay,

which represents a function with compact support in terms of its derivatives. For an elliptic operator L of order >2 with constant coefficients

and only highest order terms, one can construct explicitly a

POTENTIAL THEORETICAL APPROACH

213

function J(x) related to it in the same way as (xlz-n is related to the Laplacean A. More precisely, the function J (x) will have the following properties. (i) J (x) is a real analytic function for I x I 0. (ii) If n is odd or if n is even and n > m, then

(3)

J (X)

I(x)

where w(x) is positive homogeneous of degree 0, (w(lx) = w(x), t > 0). If n is even and n S m, then

J (x) = q(x) log

IxI +

I(x)

where q is a homogeneous polynomial of degree m - n. (iii) The function J (x) satisfies the equation

L0J (x) = b (x)

(4)

so that for every C., function with compact support (5)

f

f

0(x) = [Lo/(y)]J(x -y) dy = Lo s6(y)J(x -y) dy

The result is classical; we carry out the construction only for the case of an odd n > m and for m = 2 and all n. (As a matter of fact, we shall assume that n is odd and exceeds m whenever it is

convenient to do so during this chapter. This assumption, howdenote the characteristic form of ever, is not essential.) Let Lo, i.e., let a,,DP, I a,E" Lo = IPI =»6

IP1 =m

Consider the function

Ix = x1E1 + ... +

It is obvious that this function is positive homogeneous of degree m + 1 and real analytic for where x -

x 0 0. It is easy to see that Lol(x) is positive homogeneous of degree one and invariant with respect to any rotation about the origin. Hence LOI(x) = const. Ixt. Now set J(x) = cAn+112I(x), where c is a constant. This function has properties (i) and (ii).

L. BERS AND M. SCHECHTER

214

That it also has property (iii) for an appropriate choice of the constant c follows from the fact that L0J(x)

= Loci"+1/2I(x) = cA"+1I2LOI(x) = c0"-1/2 const. IxI = const. A(An-1I'2 IxI)

= const. A x12-" = b(x)

When m = 2 we can write down the fundamental solution just as easily. In fact if

=.I a,,$,s, let A denote the cofactor of a1, in the determinant la,,l. Then

J(x) = c(,J

A.1x,x,)2-,7/2

J (x) = c log (I A1,x,x,)

n>2 n=2

One readily verifies that these expressions satisfy the required stipulations.

The function J (x) just constructed is called a fundamental solution of the differential equation Loo = 0. More generally, for any homogeneous elliptic equation L¢ = 0, a function J (x,)r) depending on a parametery is called a fundamental solution if it satisfies the equation L.J (x,y) = b(x - y)

It should be remarked that bounds on the derivatives of w (x) in (3) depend only on n and the ellipticity constant of Lo, i.e., on

min IQ( ) IFI=1

Fundamental solutions for elliptic equations have been constructed by many authors, the most important results being due to John whose book [1] also contains an extensive bibliography. In

particular, fundamental solutions are known to exist for any equations with analytic coefficients (in which case the fundamental solution itself is analytic) and for any equation with C,o coefficients (in which case the fundamental solution is itself C.) as well as under much weaker assumptions. As a matter of fact, it

is probable that the fundamental solution can be constructed

POTENTIAL THEORETICAL APPROACH

215

for any equation with Holder continuous coefficients, though as far as we know, this has never been proved in the literature.

Fundamental solutions play an important part in the theory of elliptic equations. In particular, once a sufficiently nice fundamental solution has been constructed, one can easily prove analyticity and differentiability of solutions. An arbitrary solution

of an equation defined in a domain 9 with a sufficiently nice boundary ! can be expressed in terms of its Cauchy data at the boundary (values of the function and its derivatives up to the order m - 1) in terms of the fundamental solution of the adjoint equation. If a fundamental solution satisfies appropriate boundary conditions (in which case it is called Green's function), a solution can be expressed in terms of fewer boundary data. Fundamental solutions also play a part in the integral equation approach to boundary value problems. This method is classical for the potential equation and for second order equations in general;

an excellent account will be found in the book by Miranda [1]. The integral equations method has recently been extended to

equations of higher order (Agmon [4] and others). But our concern here will be only with the use of the simplest fundamental

solutions, namely the functions J (x) constructed above. These

applications are based on a simple device due to Korn and Lichtenstein.

To an elliptic operator of order m with variable coefficients a,(x) and to every point x0 we associate the "tangential operator"

=

v

lpl=yn

which is a homogeneous operator with constant coefficients. We shall denote by J.,°(x) the fundamental solution of the equation Lx°4 = 0 constructed above. JZ° is called a parametrix for the equation Lc = 0 with singularity at x0. We shall denote by Sx0

the operator which takes a function O(x) into the function ti = Sx00, where (7) V (x) =fix°(x -y)o(y) dy Finally, we define the operator Tx° by the equation (8)

Tx° = SX°(LZ° - L)

L. BERS AND M. SCHECHTER

216

Since S 0L, ,

= 1 (= identity) on functions with compact support

and similarly L 0ST0 = 1, by virtue of identity (5), we have the following If 0 has compact support, then

"LEMMA" A.

= Tx0 ± Sx0L¢

(9)

and if 0 = Txa¢ ± Sxu f

(10) then

is a solution of the equation Lc = f.

The quotation marks around the word lemma are in recognition

of the somewhat cavalier way in which the lemma has been stated. It goes without saying that in order to become a real lemma,

it must be supplemented by a statement on the assumptions involved. In applications to be considered, all this will be selfevident. The proof of A is clear. Equation (9) follows by noting that on

functions with compact support T,0 + S= L = 1. On the other hand, assuming that (10) holds we have (Tx0 ; Sx L)o = Tx0o -Sx0 f hence, S_L4 = S.,, ,f and Lq = L.,,, S,.0L0 = L,,Syo f =f. The applications of Lemma A are based on certain inequalities which are best stated in terms of certain norms. Some Function Spaces

5.2.

We consider first C X functions defined on the closure of a boun-

ded domain1i. For these functions we set

0,

(jIvdx)

1/r

(r > 1)

l.u.b. ICI

H..gr[o] = l.u.b. ,._0

Ix, - X "l

0 < x < 1

(diam V)' max IIDp0!Ic,(9), In!

k = 1, 2, .. .

POTENTIAL THEORETICAL APPROACH II

(diam V)k-" max H,,,,[Dp0],

IICktx() = It'IICk(l)

1PI =k

k=o,1,..., II0IIJi ()

Il

217

=

(diam

llNk(W) = I (diam q)Iv i -,nlr IPlsk

0 0) and is a C. function for x (22)

w(x) dS = 0

J IxI =1

0, and if

POTENTIAL THEORETICAL APPROACH

223

where dS denotes the area element on the unit sphere. (The smoothness condition on w can be considerably relaxed while the condition (22) of vanishing mean value is essential.) It is easily seen that (22) is equivalent to (22')

K(x) dx = 0

I

R1

C 0(1)

Applying the Calderon-Zygmund inequality to (26) we have for IPI =m (29)

Rm-nI4 IID1%11L°(R) C o(1)

For 0 < IpI < m we estimate q > n. We have that

II Dny II C,, (R)

using the assumption

I D"J (x) I C const. lxI m-n-IDI

so that for Ixi < R ID"A

const.

Irm-n-IDI

* Iv'l

I

1 /q'

< const. 11+P11L°(R)

)( 1

Ix

l IvI n) (38)

IIUIIR",(RI)
R,

where R1 is some fixed positive constant less than I xJf2. The function Ix - zIl-n is in C" for IzI < R1 and vanishes for z = 0. Hence we may differentiate under the integral sign. Since K(x -y) is in C., for Ix -yi > R1, the same is clearly true Ixll-n

for the second integral as well. This completes the proof. It is obvious from our proof how one may relax considerably the smoothness assumption on &)(x).

Appendix II.

Proofs of the Interpolation Lemmas

We now give the proofs of Lemmas 1 and 3 of Section 5.2..

Assume that 9' is contained in some cube with sides of length 2R. Since we may take Ou to have compact Proof of Lemma 1.

support, we may assume that 9' is the cube. Thus it clearly suffices to prove IIOIlc,((l) < M

L. BERS AND M. SCHECHTER

264

This estimate is stronger than the Schauder estimate of the preceding chapter since only the size but not the continuity properties of the coefficients are involved.

The proof of the estimate is based on the representation theorem and on a known theorem in function theory, Privalofj's theorem (proved in the Appendix), which states that iff (z), IzI < 1 is analytic and Ref (z) is continuous on I z I = 1 and satisfies there a Holder condition with exponent 6 < 1 and constant K, then f (z) satisfies on Izi < 1 a Holder condition with exponent 6 and constant CK, where C depends only on 6. It was noted above that the function w(z) satisfies equation (6),

i.e., an equation of the form (8) where the complex-valued coefficients u(z), . . . , y(z) are measurable functions satisfying the inequalities (9). The constants k, k', k" depend only on the uniform ellipticity of (5). Applying the representation theorem we have w(z) = e81zf[x(z)] + so(z)

(20)

Here C = x(z) is a liomcomorphism of the closed unit disc onto itself which leaves the points 0, 1 fixed; the functions s(z), so(z) are continuous in the closed unit disc, real on the unit circle, and vanish at z = 1, and the function f is regular analytic for Cl < 1 and is continuous on the unit circle. Also, we know bounds and Holder conditions for the functions s(z), so(z), x(z) as well as for the inverse homeomorphism z which depend only on the constants k, k', k". I

Since T'(e'0) is the tangential derivative of a single-valued

function it has mean value zero and hence vanishes somewhere on the unit circle, so that by hypothesis (21)

1 -r' l < 2K1,

Ki depending on the Holder continuity of T'

Also, from (17) we infer, by considering the variation of 0 on a curve of steepest descent, that there exists a point zl in the unit disc such that (22)

l ic(zi) I < K21

K. depending only on the Holder continuity of T'

FUNCTION THEORETICAI. APPROACH

265

The boundary condition (17a) may be written in the form

Re [izw(z)] = -r'(z),

(23)

IzI = 1

or, since s and so are real on Iz1 = 1, (24)

Rc {izf[x(z)]} = e-812>[T'(z) - so(z) Re (iz)],

We have that x-1 (ei°) = e;n(o)

IzI = 1

0 1, as we may suppose, Lemma 13 yields Ig(z) I < c for IzI < 1/R. Hence Ig'(z) I < 4cR for IzI S 1/2R. This implies (26) if I z11, I z21 ? 3R.

Finally, (26) is trivially fulfilled if Iz1I < 2R and Iz21 z 3R. COROLLARY.

I W"(Zl) - W"(Z2) I< C IZ1 - Z21"

This implies statement (y) of Chapter 6. We now prove statement (E). The uniqueness part follows at once from Theorem 3. To establish existence, we define u and a to be zero outside the unit disc. Then 11a II

w in the form

7r1l'k'. We represent

w=0+w"'"

We must determine 0 in such a way that Im

on IzI = 1 and

- ('01"') = 0

- co-"," = 0 at z = 1. Set /(z) = (D(W"(z)).

We must determine a holomorphic function V(z) in the unit disc such that Im tD = -r = Im co-"-" ° (W")-1 on IzI = 1 and D = w"0L

FUNC'T'ION THEORETICAL APPROACH

279

at z = 1. Note that we know Holder conditions for W'`, (W,,)-1, w'`.", and r depending only on k and k', as well as a bound on Iw'`-"(1) I. It follows from Privaloff's theorem (cf. below) that a 0 with the desired properties exists and satisfies a Holder condition depending only on k and k'. This proves (e). It remains to prove Privaloff's theorem. THEOREM 5 (Privaloff [1]). Let f (z) = u + iv be analytic in 1 and satisfies z I < 1. If u (z) is continuous in I z I u(ezas) I (27) Iu(eia,) < K Ie'e, e'0,I2

-

-

then f (z) is continuous in I z I < 1 and satisfies

If(zl) -f(z2)I 1 is a constant. The latter equation is elliptic only if the flow is subsonic, i.e., if Ol + Oy < 2/(y 1).

-

7.1.

Boundary Value Problems

In studying boundary value problems for equation (1) we shall assume, merely for the sake of simplicity, that the domain considered is the unit disc I z I < 1 (x + iy = z). We shall consider the Dirichlet problem

(6a)

= r (given function) on 1': 282

IzI = 1

QUASI-LINEAR EQUATIONS

283

and the Neumann problem which is convenient to formulate as follows :

ao = c + y (constant + given function) on I',

(6b)

0 = 0 at 1 (The constant c is not prescribed but is to be determined.) In order to establish uniqueness theorems for a boundary value

problem one usually makes use of the obvious but important observation that if the coefficients of (1) are continuously differen-

tiable, the difference co = 01 - 02 of two solutions satisfies a linear elliptic equation. In fact, we have that i = 1, 2

A1o:,xx + 2B,0t>xv + CA,vv + D, = 0,

where A; = A(xy,0t,0i,x,0t,,,), etc. Subtracting the two equations, we obtain that (7)

A2(k)x, + 2B2wxv + C2wvv + (A2

- A1) 01,

+ 2(B2 - B1)01,xv + (C2 - C1)01,vv + (D2 - D1) = 0. The differences (A2 - A1), ... , can be computed as follows. Set

F(t) = A[x,y, 02 + (1 - t)01, 02,x + (1 - t)41,x, 02,v + (1 - t)01,vl Then

dt + to,, f

A2 - Al = fol F' (t) dt = co f lA,6 dt + (1,S f

dt o1A,.

o

Thus (7) may be written in the form (8)

A2w= + 2B2wxv + C2wyy + awt + bw + cw = 0

where 1

a = 01, faAd= dt + 201, fo

1

1

dt + 01.vv oCo= di ±

1

o

D,= dt

and b and c are defined similarly. Uniqueness of the Dirichlet or Neumann problem will be established if we can assert that the

284

L. BERS AND M. SCHECHTER

linear equation (8) has only the trivial solution u.) = 0 under the homogeneous boundary conditions. This is so, for instance, if A, B, C do not depend on 0 and aD/ao S 0. In this case c < 0 and the maximum principle is applicable (cf. Chapter 2).

7.2.

Methods of Solution

Existence proofs for our two boundary value problems can be reduced to the search of a fixed point of a transformation in an appropriately chosen Banach space B, the elements of which are functions defined in the unit disc. Let 'D(x,y) be an element of B. We form the linear equation (9)

airy + 2box, ± co,, = d

where

a(x,y) = A(x,y,t,(x,(D), .. .

and solve for it the boundary value problem considered. (We assume, of course, that this is possible, and in a unique way.) Denote the solution by ¢ = TO and assume that it belongs to B. Then T is a (nonlinear) mapping of B into itself and a fixed point

of T, i.e., a 0 such that 0 = To is a solution of the nonlinear boundary value problem. We will now describe several methods for proving the existence of a fixed point of T or, what is the same, of a root of the equation (10)

K(D=(1-T)(D =0

(a) Successive Approximations.

One may try to find a fixed point

of T by choosing some elements 0 and forming successively 01 = T O O , D2 = T01i etc. If the method converges, that is if (D)

'V and if T is continuous, then TO = (D. The convergence of the method of successive approximations

is often assured by the so-called principle of contracting mappings :

If T maps a closed convex subsets of a Banach space B into itself, and if there exists a constant 0, 0 < 0 < I such that (11)

IIT'V1 - T'V211 0). We also assume that the boundary functions T and v in (6) are of class Cl_, and C2, respectively. Finally, and for the sake of brevity,

we assume that A, B, C are Holder continuous in their five

QUASI-LINEAR EQUATIONS

287

variables, though mere continuity would suffice (cf. BersNirenberg [2]. For an application of the same idea to gas dynamics see Bers [6]).

For B we take the space Ci+a (1), fi to be determined later. If 0 e C,.,_,,(1) and we form the linear equation

A(x,y,O,(Dx,O.)0zz + 2B(... ) 0., + C( ... )0y, = 0 then, according to the results of Chapter 6, our Dirichlet problem (or our Neumann problem) will have a unique solution 0 = TO. If we choose y sufficiently small we will have (14)

(15)

II0 II c,+Y

=

11TO ll el+V

1, we have to put x. Put H2 = L2(R,a,v) where da = dt on N and 0 otherwise. Finally, define a mapping V from Hl to H2 by putting (8)

(VF)k(t) =f

;F(E)hk(t,E)

when t e N and VF(t) = 0 otherwise. The reader will have no trouble verifying that V is unitary and that W = VU, which is a unitary transformation from H to H2i diagonalizes B canonically. An easy example is furnished by or, in quantum mechanical terms, the nonrelativistic Hamiltonian of n free particles. Observe that the spectrum of any B of the form (7) has uniform infinite multiplicity when n > 1. From (8) we obtain an analogue of (0.2) by expressing F = Uf in terms off. Formally the result is where (9)

( 4Tf)k(t) = (f,4k(t))

Ok(t,x) =Jet

`hk(t,)

- dwt()

EIGENFUNCTION EXPANSIONS

311

If Se is compact, which it is if B is elliptic in the sense that the polynomial p has definite principal part, then S6k is an eigenfunction

of B, but it is not square integrable. When St is not compact, but the functions h have compact supports, we still get functions

0, but for arbitrary orthonormal sets this is no longer true. To get an example of this, we can, e.g., put n = 2 and p = i8E2 with a larger integer s. Parameterizing St with dcul =

we get

28 d$j

so that any function vanishing at infinity and behaving like for small values of $, is square integrable on St. For such a function, however, the right side of (9) is not defined, but it is easy to see that it makes sense as a distribution. In fact, we shall see later that the eigenfunctions of differential operators always exist as distributions. III. Generalized Eggenfunctions

We are going to study situations when the formula (0.2) degenerates without losing its sense altogether. We begin with an interesting half-way abstract case treated by Gelfand and Kostyucenko [8]. First, a few notations. We let S = S(V) be the set of all complex infinitely differentiable functions vanishing outside compact subsets of an open subset V of real n-space. The results are true

also for functions with values in a finite-dimensional complex space, but we stick to complex functions for simplicity. Derivatives

will be denoted by Dk = a/iaxk and we put

D. = Dal, ... , D«,n and denote its order by Ial = m. To these symbols we add Da = 1, of order 101 = 0 and the norms max IDJ(x)I,

1«I < m

and

IDmfI = sup IDmf(x)I,

xeV

A subset W of V is called precompact if its closure W is compact and contained in V. We let S' be the space of distributions in V

(see [21]). It consists of all linear functionals L on S with the

L. GARDING

312

following continuity property. To every precompact open subset V' of V there exists an integer m and a constant c such that

IL(f)I n

It is p - n - 1 times es continuously differentiable and it is a fundamental solution of the differentiable operator

L = L (D) = Y

DX,

17.1

p

in the sense that

fC(x -y)Lf(y) dy =f(x) for any f in S(R). Let V' and V" be open precompact subsets of V which telescope so that V' c V" c: 17" c V, let h c- S be 1 on V' and put

b(x,y) = h(x)C(x -y)h(y) Then, if f E S(V') we have (3)

f

f (x) = b(x,y)Lf(y) dy

It is clear that if we choose p large enough, then b(-,_y) E H

for every}', and an easy approximation argument will show that operator L' commutes with the integration in (3) so that (4)

fB(E,y)Lf(y)

dy

L. GARDING

314

for almost all , where (Ub(-,y))($)

This is true in fact for any bounded operator from H to Hl. It is not difficult to prove that the function B can be chosen to be a Borel function in both arguments. Since U is unitary, Ib(-,y)12 =

f

da(d)

so that

j'Ib(.,.y)I2d y=f

da(d) dy
0 such that ft(x)c2(x) dot (x) < oo

Stripped of measure-theoretical detail, the proof of this theorem is very simple. From (8) we conclude that

f

(B*f) (y) = b(y,x)f(x) do, (x)

where f c C(B). It is not difficult to see that the operator U commutes with the integration and this gives (UB*f)(A)

=f B(1.,x)f(x) d«(x)

where B(A,x) = (Ub(x, -))(A)

(12)

Dividing by h we get (10) with

4(A,x) = h-'().)B(2,x)

(13)

It is easy to prove that the components of O(2) are linearly independent (see [2]). To get (11), we combine (12) and Parseval's formula, getting

f

c2(x) = I b(x,y) I2 dx(y) = f IB(2,x) I2 dd(.)

so that multiplying by t(x) and integrating

f

ff IB(2,x)12 t(x) dx(x) da(A) = t(x)c2(x) - dx(x) < oo

By virtue of (13), this implies (11) for almost all A.

Clearly, Theorem 2 applies where A itself has the Carleman property and this makes it possible to apply it to any self-adjoint operator A on a separable Hilbert space H. In fact, if we refer

L. GARDING

318

H to a complete orthonormal set we can think of it as a space of functions L2(W,x) where W consists of all integers of arbitrary sign and y assigns the measure one to every integer. If we choose all elements g1, x e W, of the orthonormal set in D(A), then where

f c D(A)

(Af) (x) = I a(x,y)f (y), a(x,y) =

is the matrix of A. Hence, A has the Carleman property, (9) reduces to

I If(x) I IAg.I < oo we have

and

A (A,x) = (Ua(x, ))(A)

(14)

I0().,x)I2

t(x)

< Co

where now t(x) IAgsI2
0 and

+. + a L* = Dma. m

0

its adjoint with respect to the scalar product (1)

(f,g) = f fg dx

We require that L be formally self-adjoint, i.e., that L* = L. The coefficients of L cannot be arbitrary. We require that at has locally square integrable derivatives of order 1. We call attention here to an obvious gap in the existent general theory. Reference in the sequal were chosen more or less at random to illustrate subject, without considerations of priority or even best results. I.

Direct Hilbert Space Approach to Solutions of the Heat Equation

We describe in this section a proof (based on a method of Vishik') of the existence of solutions of (I-1)

-Au +

at =f(X,t)

1 M. I. Vishik, Mixed boundary problems, Dokl. Akad. A'auk SSSR, 97, 193-6 (1954). 329

A. N. MILGRAM

330

where

0=

a2

axe

+

...

+

a2

and

x = (x1, ... , x,,)

a n,

0 0 for each E > 0.

fP -Pol ? e

3 Sternberg, Math. Ann., 101, (1929).

4 I. Petrowsky, Zur ersten randwertaufgabe der warmeleitungsgleichung, Compositio Math., 1 (1935). b This definition is contained in an unpublished paper of Professor W. Fulks, Department of Applied Math., University of Minnesota. See also, J. L. Doob,

A probablity approach to the heat equation, Trans. Amer. Soc., 80 (1955).

A. N. MILGRAM

336

The solution to the boundary value problem is sought as the infimuin of supcrparabolic functions with boundary values > a given function]. In his treatment of the above problem, Petrowsky restricted himself to the case of one space dimension: a2u/ax2 = au/at, and considered a domain bounded by two curves

x=0i(t)

X = 02(t),

0 - . Such refined conditions have not been obtained for the higher dimensional case. A crude sufficient condition for the existence of a barrier at a point PO E D5 is that PO be the vertex of a cone in the complement of D along the axis of which the t coordinate of a point decreases as the point recedes from PO (see figure above).

The problems discussed in the preceding sections have one feature in common in that all seek solutions of Au = U= which assume prescribed boundary values on D the boundary of a general domain in space-time. The solution sought cannot in general achieve all prescribed boundary values. For example, if IV.

PARABOLIC EQUATIONS

337

D is a cylinder with axis parallel to the t axis, the boundary values assigned at the top of the cylinder will not influence the solution

if the solution conforms to preassigned values on the lower boundary. Thus the boundary value problems considered must be regarded as a search for solutions of Au = u, which in some sense minimizes the difference between a prescribed function f and the values assumed by u on D. The word "minimize" as used in Section III means equal on a boundary set of "greatest measure." But if minimize were to mean the solution u(p) = u(x,t) is sought

so as to minimize the integral f If(p) - u(p)12 dS taken over the boundary, a new class of problems would arise. In such a problem prescribed boundary values on the top of a cylindrical domain would have material influence on the solution. A question related to the above which might be asked is the

following: let R be a subdomain of D. Let f be a prescribed function on R the boundary of R. What boundary values should be assigned on aD so that the solution u(x,t) best approximates f

on aR ? We leave open the sense in which the words "best approximates" are used. Of the methods used in Sections I and II we can say that the first is modern, the second classical. In the direct Hilbert space approach a solution of the boundary value problem is obtained by first solving Au - au/at = f with 0 boundary values. Then using f = - Af0 - afol at, where fo is a function defined and belonging to C2 in D, the function v = u + f0 provides a solution of Av aav/at with prescribed boundary values. The listener will observe at once that this seems to limit the boundary values to those which admit an extension into the interior of D of class C2(D). This would indeed be a very serious defect in the Hilbert space approach. Fortunately, in the case of second order equations the maximum principle comes to the rescue by guaranteeing that a sequence of solutions with uniformly convergent boundary values is uniformly convergent in the interior and the limit is readily shown to be a

solution. Hence the fact that the Hilbert space method is not directly applicable to all preassigned boundary values is not of vital importance. For equations of order higher than 2 the situation is different.

A. N. MILGRAM

338

V.

Equations of Higher Order

Let D be a domain in E" x J, and a = Jti, a2, ... , xK a sequence of integers where 1 < a, < n for each v. Set lal = K $12, ... , 41K. The and D" = aK/(axai 4,2 . . . 4,K), equation

a xt D"u =

V-1 jaj:z_ 21n

"

au at

will be called parabolic if Z a,D" is a strongly elliptic operator, i.e., if there exists a constant A > 0 such that

Rt[,ja.(x,t)e] z (-1),,1+1A

for2, ... ," real and

I

IEI2"

I2 = ($1)2 + ...

($1)2

0,

and (x,t) E D. Systems of equations can also be treated; definitions will be found in the other lectures. Various methods for treating

(V-1) have been given. The most direct and comprehensive is probably the method described in Section I, suitably modified. Of course, the solution obtained is at first only known to be a weak solution, this difficulty is easily overcome. Using the fundamental

solution whose existence and regularity was established (for systems) by Eidelman6 it is easy to verify by known methods that the weak solution is regular. Other methods including Friedrich's mollifiers can be used to verify regularity of the weak solutions in the interior. However the requirement that the boundary values can be extended into the interior, i.e., assumed by a function fo

which has finite m-norm, cannot this time be eliminated by a simple limiting process. For higher order equations no maximum principle is known. Thus even for elliptic equations the general existence theory based on Hilbert space methods is inadequate

to prove, say, the well-known theorem that the biharmonic equation L2u = 0 admits a solution u in the interior of a sphere so

that u = g, au/ an =f for arbitrary continuous functions f and g on the boundary. It is likely that at least for sufficiently smooth boundaries, the method analogous to that discussed in Section II will enable us to 6 S. D. Eidelman, On the fundamental solution of parabolic systems, Mat. Sb. N.S., 38, 80 (1956).

PARABOLIC EQUATIONS

339

obtain more complete results. Recently R. Juberg (Thesis, University of Minnesota) observing that for the fundamental solution (Eidelman) similar jump relations continue to hold, derived integral equations analogous to those described in Section II. The resulting system of equations, however, are complicated,

and his results are definitive in certain special cases. It is clear that an investigation of the integral equations both for parabolic and elliptic equations in an effort to obtain a Poisson integral type representation of the solution must lead to new and deeper insight into the boundary value problem for parabolic and elliptic equations.

Index

A priori estimates, 231, 263, 286, 289 Adjoint operator, 138 Analyticity theorem, 136, 139, 207

Direct integrals, 306 Dirichiet and Neumann problems, 263, 286

Dirichiet principle, 164 Dirichiet problem. 141, 152, 153, 154, 155, 159, 190, 196, 197, 237, 263, 282, 329 for the difference equation, 157 Discontinuities, propagation of, 53 Distribution, 138 Duhamel's integral, 17, 63

Banach-Saks theorem, 181 Beltrami equation, 257, 267 Boundary conditions, 141 Bicharacteristics, 58 Bounded linear functional, 179

Calderon-Zygmund inequality, 224, 245

Cauchy and Kowalewski theorem, 46 Cauchy data, 44, 62 Cauchy problem, 18, 45 standard, 63 uniqueness of, 47 Cauchy-Riemann system, 257 Characteristic form, 43, 70, 135 Characteristic lines, 7 Characteristic matrix, 43 Characteristic surface, 21, 43 Classical solution, 137 Closed linear subspace, 178

Elastic waves, equation of, 74 Elliptic, 143 Elliptic equations, 134, 135 Elliptic operators, 143 Elliptic systems, 255 Energy integrals, 24

Coerciveness, 201

Compact support, 7 Complete, 178 Completely continuous, 183,199 Complex notations, 256 Continuity method, 238, 285

Finite differences, solution of parabolic equations, 109 stability of difference schemes, 115 Fourier transformation, 65 Fredholm pair, 142, 175 Fredholm-Riesz-Schauder theory, 183

Descent, method of, 16

Free surfaces, 43 Function spaces, 216

Differentiability theorem, 138,

Estimates "up to the boundary," 235 Finite differences, 155 Finite differences, method of, 108

see also Symmetric hyperbolic systems

Fundamental identity, 211 Fundamental solution, 214, 215, 333

139,

174 341

INDEX

342 GArding's inequality, 170, 198

Liebmann's method, 159

General boundary value problems,

Linear equations with wave operator as principal part, 31

142

Generalized derivatives, 138 Generalized eigenfunctions, 305, 311 Gradient vector, 38 Green's function, 215 Green's identity, 48, 104

Heat

conduction,

boundary initial

value problem for rectangle, 104 equation of, 94 initial value problem, 98 maximum principle for, 96, 97 smoothness of solutions, 101 uniqueness of solutions, 96, 97 Heat equation, 329 Hilbert space, 177 Hilbert space approach, 329 Hilbert space methods, 165 Holder condition, 136 Holder continuity of derivatives, 136 Holder continuous, 138 Holder- Korn-L ich tenstein-Giraud inequality, 223, 244 Holmgren, uniqueness theorem of, 47 Huygens' principle in strong form, 15 Hyperbolic equations, linear, 69 solution by plane waves, 72 standard Cauchy problem for, 63 wave equation, 1, 2

Hyperbolic systems, Green's identity for, 48 with constant coefficients, 70

see also Symmetric hyperbolic systems

Integral equation approach, 215 Integral equations, 329 Interior estimates, 232 Interior regularity, 190 Interpolation lemmas, 218, 219, 250 Laplace equation, 134, 136 Lax-Milgram lemma, 180, 199 Leray-Schauder method, 285

Lions, 202

Maximum principle, 150, 160, 262, 334

Maximum principle for heat equation, 96

Maxwell's equations, 87 Minimal surfaces, 282 Mixed boundary problem, 143 Mollifiers, 145

Morrey-Nirenberg, 210 µ-conformal mapping, 267 Multi-index, 135 Negative norms, 200 v. Neumann condition, 118 Neumann problem, 154, 283 Norm, 135 Normal boundary conditions, 142 Normal velocity, 23,61

Oblique derivative problem, 154 Ordinary differential operators, 319 Parabolic equations, 2, 94, 329

see also Finite differences and Heat conduction equation Parseval's identity, 65 Periodic distribution, 168, 169, 174 Perron's method, 329, 334 Plane waves, decomposition into, 72 Poisson equation, 134, 137 Potential gas flow, 282 Potentials, 333 Potential theory, 211

Principle of contracting mappings, 283

Privaloff's theorem, 264, 267, 279 Projection theorem, 164, 177, 179 Propagation of discontinuities, 53 Properly elliptic, 144

Quasiconformal, 258

343

INDEX Regularity at the boundary, 200

Representation theorem, 164,

179,

259

Riesz-Schauder theory, 164 Runge property, 140

Scalar product, 178 Schauder estimates, 231, 238 Schauder's fixed point theorem, 286 Schwarz inequality, 166 Second-order equations, 150 Self-adjoint operators, 303, 305 Single hyperbolic equations, 62 Singular kernel, 222 Smoothness of strong solutions, 240 Sobolev and Kondrashov, 221 Sobolev's inequality, 220, 242 Sobolev's lemma, 30, 77, 167 Spaces Ht, 165, 167 Spacelike surface, 23

Spectral theorem, 303, 304, 305, 306, 307, 308 Stability of difference schemes, 116 Strictly hyperbolic, 72 Strong derivative, 137 Strong solution, 138 Strongly elliptic, 143, 144 Sturm-Liouville operators, 319 Subspace, 178

Support of a function, 7 Symmetric hyperbolic systems, 86, 87 difference methods for, 117

Tangential operator, 215 Test functions, 137 Timelike surfaces, 23 Titchmarsh-Kodaira formula, 320

Uniformly elliptic, 150, 255 Unique continuation, 162 Unique continuation property, 140 Unique continuation theorem, 262

Uniqueness, Holmgren's theorem of, 45

Uniqueness for heat equation, 97 Uniqueness theorems, 283

Wave equation, d'Alembert's solution for, 5 characteristics, 21 characteristic cone, 21 characteristic lines, 7 characteristic surface, 21

domain of dependence, 7 Huygens' principle in strong form, 15 inhomogeneous, 17, 18, 19 initial value problem for, 6, 10 mixed problems for, 35

mixed problem for finite x interval, 9

mixed problem for semi-infinite x interval, 7 one-dimensional, 4 spacelike surfaces, 21 three-dimensional, 10 timelike surfaces, 23 two-dimensional, see Descent, method of

Waves, see Plane waves and Elastic waves

Weak derivative, 137 Weak equals strong, 144 Weak solution, 138, 198 Weak vs. strong, 137 Well-posed problems, 2

Zeros of elliptic systems, 261

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