Introduction to Partial Differential Equations: Second Edition 9780691213033

Citation preview

INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS

INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS SECOND EDITION

GERALD B. FOLLAND

PRINCETON UNIVERSITY PRESS • PRINCETON, NEW JERSEY

Copyright © 1995 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved Library of Congress Cataloging-in-Publication Data Folland, G. B. Introduction to partial differential equations / Gerald B. Folland.— 2nd ed. p. cm. Includes bibliographical references and indexes. ISBN-13: 978-0-691-04361-6 1SBN-10: 0-691-04361-2 1. Differential equations, Partial. I. Title. QA374.F54 1995 95-32308 515\353—dc20 The publisher would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton Academic Press 10 9 8 7 6 5 4 3 2 1

CONTENTS

PREFACE Chapter 0 PRELIMINARIES A. Notations and Definitions B. Results from Advanced Calculus C. Convolutions D. The Fourier Transform E. Distributions F. Compact Operators Chapter 1 LOCAL EXISTENCE THEORY A. Basic Concepts B. Real First Order Equations C. The General Cauchy Problem D. The Cauchy-Kowalevski Theorem E. Local Solvability: the Lewy Example F. Constant-Coefficient Operators: Fundamental Solutions Chapter 2 THE LAPLACE OPERATOR A. Symmetry Properties of the Laplacian B. Basic Properties of Harmonic Functions C. The Fundamental Solution D. The Dirichlet and Neumann Problems E. The Green's Function F. Dirichlet's Principle G. The Dirichlet Problem in a Half-Space H. The Dirichlet Problem in a Ball I. More about Harmonic Functions

ix 1 1 4

9 14 17 22 30 30 34 42 46

56 59 66 67 68 74 83 85 88 91 95 110

vi

Contents

Chapter 3 LAYER POTENTIALS A. The Setup B. Integral Operators C. Double Layer Potentials D. Single Layer Potentials E. Solution of the Problems F. Further Remarks

116 116 120 123 129 134 139

Chapter 4 THE HEAT OPERATOR A. The Gaussian Kernel B. Functions of the Laplacian C. The Heat Equation in Bounded Domains

142 143 149 155

Chapter 5 THE WAVE OPERATOR A. The Cauchy Problem B. Solution of the Cauchy Problem C. The Inhomogeneous Equation D. Fourier Analysis of the Wave Operator E. The Wave Equation in Bounded Domains F. The Radon Transform

159 160 165 174 177 183 185

Chapter 6 THE L 2 THEORY OF DERIVATIVES A. Sobolev Spaces on M" B. Further Results on Sobolev Spaces C. Local Regularity of Elliptic Operators D. Constant-Coefficient Hypoelliptic Operators E. Sobolev Spaces on Bounded Domains

190 190 200 210 215 220

Chapter 7 ELLIPTIC BOUNDARY VALUE PROBLEMS A. Strong Ellipticity B. On Integration by Parts C. Dirichlet Forms and Boundary Conditions D. The Coercive Estimate E. Existence, Uniqueness, and Eigenvalues F. Regularity at the Boundary: the Second Order Case G. Further Results and Techniques H. Epilogue: the Return of the Green's Function

228 228 231 237 242 248 253 261 263

Contents

vii

Chapter 8 PSEUDODIFFERENTIAL OPERATORS A. Basic Definitions and Properties B. Kernels of Pseudodifferential Operators C. Asymptotic Expansions of Symbols D. Amplitudes, Adjoints, and Products E. Sobolev Estimates F. Elliptic Operators G. Introduction to Microlocal Analysis H. Change of Coordinates

266 266 271 279 283 295 297 304 310

BIBLIOGRAPHY

317

INDEX OF SYMBOLS

320

INDEX

322

PREFACE

In 1975 I gave a course in partial differential equations (PDE) at the University of Washington to an audience consisting of graduate students who had taken the standard first-year analysis courses but who had little background in PDE. Accordingly, it focused on basic classical results in PDE but aimed in the direction of the recent developments and made fairly free use of the techniques of real and complex analysis. The roughly polished notes for that course constituted the first edition of this book, which has enjoyed some success for the past two decades as a "modern" introduction to PDE. From time to time, however, my conscience has nagged me to make some revisions — to clean some things up, add more exercises, and include some material on pseudodifferential operators. Meanwhile, in 1981 I gave another course in Fourier methods in PDE for the Programme in Applications of Mathematics at the Tata Institute for Fundamental Research in Bangalore, the notes for which were published in the Tata Lectures series under the title Lectures on Partial Differential Equations. They included applications of Fourier analysis to the study of constant coefficient equations (especially the Laplace, heat, and wave equations) and an introduction to pseudodifferential operators and CalderonZygmund singular integral operators. These notes were found useful by a number of people, but they went out of print after a few years. Out of all this has emerged the present book. Its intended audience is the same as that of the first edition: students who are conversant with real analysis (the Lebesgue integral, Lp spaces, rudiments of Banach and Hilbert space theory), basic complex analysis (power series and contour integrals), and the big theorems of advanced calculus (the divergence theorem, the implicit function theorem, etc.). Its aim is also the same as that of the first edition: to present some basic classical results in a modern setting and to develop some aspects of the newer theory to a point where the student

x

Preface

will be equipped to read more advanced treatises. It consists essentially of the union of the first edition and the Tata notes, with the omission of the IP theory of singular integrals (for which the reader is referred to Stein's classic book [45]) and the addition of quite a few exercises. Apart from the exercises, the main substantive changes from the first edition to this one are as follows. • § IF has been expanded to include the full Malgrange-Ehrenpreis theorem and the relation between smoothness of fundamental solutions and hypoellipticity, which simplifies the discussion at a few later points. • Chapter 2 now begins with a brief new section on symmetry properties of the Laplacian. • The discussion of the equation AM = / in §2C (formerly §2B) has been expanded to include the full Holder regularity theorem (and, as a byproduct, the continuity of singular integrals on Holder spaces). • The solution of the Dirichlet problem in a half-space (§2G) is now done in a way more closely related to the preceding sections, and the Fourieranalytic derivation has been moved to §4B. o I have corrected a serious error in the treatment of the two-dimensional case in §3E. I am indebted to Leon Greenberg for sending me an analysis of the error and suggesting Proposition (3.36b) as a way to fix it. o The discussion of functions of the Laplacian in the old §4A has been expanded and given its own section, §4B. • Chapter 5 contains a new section (§5D) on the Fourier analysis of the wave equation. • The first section of Chapter 6 has been split in two and expanded to include the interpolation theorem for operators on Sobolev spaces and the local coordinate invariance of Sobolev spaces. • A new section (§6D) has been added to present Hormander's characterization of hypoelliptic operators with constant coefficients. • Chapter 8, on pseudodifferential operators, is entirely new. In addition to these items, I have done a fair amount of rewriting in order to improve the exposition. I have also made a few changes in notation — most notably, the substitution of (/ | g) for (/, g) to denote the Hermitian inner product / fg, as distinguished from the bilinear pairing (/, g) — f fg. (I have sworn off using parentheses, perhaps the most overworked symbols in mathematics, to denote inner products.) I call the reader's attention to the existence of an index of symbols as well as a regular index at the back of the book.

Preface

xi

The bias toward elliptic equations in the first edition is equally evident here. I feel a little guilty about not including more on hyperbolic equations, but that is a subject for another book by another author. The discussions of elliptic regularity in §6C and §7F and of Garding's inequality in §7D may look a little old-fashioned now, as the machinery of pseudodifferential operators has come to be accepted as the "right" way to obtain these results. Indeed, I rederive (and generalize) Garding's inequality and the local regularity theorem by this method in §8F. However, I think the "low-tech" arguments in the earlier sections are also worth retaining. They provide the quickest proofs when one starts from scratch, and they show that the results are really of a fairly elementary nature. I have revised and updated the bibliography, but it remains rather short and quite unscholarly. Wherever possible, I have preferred to give references to expository books and articles rather than to research papers, of which only a few are cited. In the preface to the first edition I expressed my gratitude to my teachers J. J. Kohn and E. M. Stein, who influenced my point of view on much of the material contained therein. The same sentiment applies equally to the present work. Gerald B. Folland Seattle, March 1995

Chapter 0 PRELIMINARIES

The purpose of this chapter is to fix some terminology that will be used throughout the book, and to present a few analytical tools which are not included in the prerequisites. It is intended mainly as a reference rather than as a systematic text.

A. Notations and Definitions Points and sets in Euclidean space M will denote the real numbers, C the complex numbers. We will be working in 1 " , and n will always denote the dimension. Points in JR" will generally be denoted by x,yt£,T]; the coordinates of x are ( x i , . . . , xn). Occasionally x\, £2, • • • will denote a sequence of points in Mn rather than coordinates, but this will always be clear from the context. Once in a while there will be some confusion as to whether (x\,..., xn) denotes a point in 1 " or the n-tuple of coordinate functions on M". However, it would be too troublesome to adopt systematically a more precise notation; readers should consider themselves warned that this ambiguity will arise when we consider coordinate systems other than the standard one. If U is a subset of Mn, U will denote its closure and dU its boundary. The word domain will be used to mean an open set Q C Mn, not necessarily connected, such that as above, the vector V(x) is perpendicular to 5 at x for every x 6 S f~i V. We shall always suppose that S is oriented, that is, that we have made a choice of unit vector i>(z) for each x € S, varying continuously with x, which is perpendicular to S at x. v(x) will be called the normal to S at ,T; clearly on 5 fl V we have

Thus v is a C function on S. If S is the boundary of a domain fi, we always choose the orientation so that u points out of f2. If u is a differentiate function defined near S, we can then define the

normal derivative of u on 5 by dvu = v • Vu. We pause to compute the normal derivative on the sphere ST(\j). Since lines through the center of a sphere are perpendicular to the sphere, we have

(0.1)

v(x) =

lZ]Lt

dv =

lf^(Xj

_ yj)dj

onS r (y).

We will use the following proposition several times in the sequel:

6

Chapter 0

(0.2) Proposition. Let S be a compact oriented hypersurface of class Ck, k > 2. There is a neighborhood V of S in M" and a number e > 0 such that the map F{x,t) = x is a Ck~1 diffeomorphism ofS x (—e, e) onto V. Proof (sketch): F is clearly Ck~l. Moreover, for each x 6 S its Jacobian matrix (with respect to local coordinates on S x IR) at (x, 0) is nonsingular since v is normal to S. Hence by the inverse mapping theorem, F can be inverted on a neighborhood Wx of each (x,0) to yield a Ck~1 map F-1:Wx^(SnWx)x(-ex,ex) for some ex > 0. Since S is compact, we can choose {xj}^ C S such that the WXi cover 5, and the maps F~l patch together to yield a Ck~x inverse of F from a neighborhood V of S to S x (—e, e) where e = min, eSj.. I The neighborhood V in Proposition (0.2) is called a tubular neighborhood of 5. It will be convenient to extend the definition of the normal derivative to the whole tubular neighborhood. Namely, if u is a differentiable function on V, for x £ S and — e < t < e we set (0.3)

dvu{x + tv{x)) - u{x) • Vu{x + tu(x)).

If F = (Fi,..., Fn) is a differentiable vector field on a subset offfi",its divergence is the function

With this terminology, we can state the form of the general Stokes formula that we shall need. (0.4) The Divergence Theorem. Let Q C 1 " be a bounded domain with C1 boundary S = —n, and it is integrable outside a neighborhood of 0 if and only if A < -n. As another application of polar coordinates, we can compute what is probably the most important definite integral in mathematics:

(0.6) Proposition. 2

Proof: Let In = fRn e'^^2 dx. Since e'*^ = H" e'***', Fubini's theorem shows that /„ = (7i)", or equivalently that /„ = {h)n^2- But in polar coordinates, i*2w

I2=

Jo

/-oo

/ Jo

e-*r\drde

/«oo

= 2* I Jo

re~r'dr

/-oo

e~*° ds - 1.

=*

I

Jo

This trick works because we know that the measure of 5i(0) in M2 is 2ir. But now we can turn it around to compute the area wn of 5i(0) in Mn for any n. Recall that the gamma function T(s) is defined for Res > 0 by /•OO

r(«)= / Jo

e~Hs-ldt.

8

Chapter 0

One easily verifies that

(The first formula is obtained by integration by parts, and the last one reduces to (0.6) by a change of variable.) Hence, if k is a positive integer, r(* + i ) = (k - I)(t - | ) . . .

T(k) = (k - 1)!, (0.7) Proposition. The area of 5i(0) in Kn is

Proof: We integrate e"^'1' in polar coordinates and set s = ?rr2:

1= [e-wW'dx= f J

=

[ e-'^r"-1 dr da

Js1(o)Jo

27T"/ 2

'

Note that, despite appearances, u>n is always a rational multiple of an integer power of w. (0.8) Corollary. The volume of 5i(0) in 1 " is

n

nr(n/2)"

Proof: / B [ ( o ) dx = wn ^ rn~l dr = ujn/n.

I

(0.9) Corollary. For any x 6 Mn and any r > 0, the area of Sr(x) is r"~ J w n and the volume of Br(x) is rnujn/n.

Preliminaries

9

C. Convolutions We begin with a general theorem about integral operators on a measure space (X,fj.) which deserves to be more widely known than it is. In our applications, X will be either M" or a smooth hypersurface in ffi". (0.10) Generalized Young's Inequality. Let (X,/x) be a a-finite measure space, and let 1 < p < oo and C > 0. Suppose K is a measurable function on X x X such that sup / \K(x, y)\ dfi(y) < C, xexJx

sup / \K(x, y)\dfi{x) < C. ytxJx

If f € Lp{X), the function Tf defined by

Tf(x)= I K(Xly)f(y)d^y) Jx is well-defined almost everywhere and is in LP(X), and ||T/|| p < C||/|| p . Proof: Suppose 1 < p < oo, and let q be the conjugate exponent (P"1 + q~l = 1)- Then by Holder's inequality,

\Tf(x)\ < ]J \K(x, y)\ d/i(y)j ' y \K(x, y)\\f{y)\" C1/9 r

/

L/ l ^ > l ! ( ) r

J

l1//p

()J

Raising both sides to the p-th power and integrating, we see by Fubini's theorem that / / x Jx or, taking pth roots,

These estimates imply, in particular, that the integral defining Tf(x) converges absolutely a.e., so the theorem is proved for the case 1 < p < oo. The case p = 1 is similar but easier and requires only the hypothesis J \K(x,y)\dfi(x) < C, and the case p = oo is trivial and requires only the hypothesis / \K{x, y)\dji{y) < C. I

10

Chapter 0

In what follows, when we say Lp we shall mean LP(IR") unless another space is specified. Let / and g be locally integrable functions on M". The convolution f * g of / and g is defined by

/ * 9(x) =

f(x - y)g(y) dy =

f(y)g(x -y)dy = g* f(x),

provided that the integrals in question exist. (The two integrals are equal by the change of variable y —> x — y.) The basic theorem on the existence of convolutions is the following:

(0.11) Young's Inequality. Iff e l 1 andgZL? (1 < p < ooj, then f*g € V and ||/* ff || p < ||/||i||ff||p. Proof: Apply (0.10) with X = K" and K(x, y) = f(x - y).

I

Remark: It is obvious from Holder's inequality that if / £ Lq and g e LP where p - 1 + q'1 = 1 then / * g 6 L°° and \\f * g^ < ||/||,||fl|| p . From the Riesz-Thorin interpolation theorem (see Folland [14]) one can then deduce the following generalization of Young's inequality: Suppose 1 < P,1,r < oo and p'1 + q~l = r " 1 + 1. If f € Li and g £ Lp then f*geLr and\\f*g\\r af in the Lp norm as e —• 0. If f G L°° and f is uniformly continuous on a set V, then f * (j>( —> af uniformly on V a s e - + 0 . Proof: By the change of variable x —> ex we see that f (f>t(x) dx — a for all e > 0. Hence,

f*Ux)-af{x) = J[f(x-y)-f(x)}$e(y)dy = J[f(x-cy)-f(xMy)dy. If / G Lp and p < oo, we apply the triangle inequality for integrals (Minkowski's inequality; see Folland [14]) to obtain

\\f*tt-af\\p 0 for each y, by Lemma (0.12). The desired result therefore follows from the dominated convergence theorem. On the other hand, suppose / G L°° and / is uniformly continuous on V. Given 6 > 0, choose a compact set W so that Jffin, w \\ < 8. Then

sup |/ * t(x) - af(x)\
G i 1 and / {x) dx = 1, the family of functions {0f}€>o defined in Theorem (0.13) is called an approximation to the identity. What makes these useful is that by choosing 4> appropriately we can get the functions f * e to have nice properties. In particular: (0.14) T h e o r e m . Iff£Lp (1 £ C%° with J — 1, and define e as in Theorem (0.13). If / G LP has compact support, it follows from (0.14) and (0.15) that / * (f>c G C%° and from (0.13) that f*e—>f'm the U norm. But U functions with compact support are dense in Lp, so we are done. I Another useful construction is the following:

Preliminaries

13

(0.17) Theorem. Suppose V C K n is compact, Q C M" is open, and V C fi. Then there exists f e Cc°°(fi) such that f = 1 on V and 0 < / < 1 everywhere. Proof: Let S - inf{|x - y\ : x € V, y £ Q}. (If Q = ffi", let 6 = 1.) By our assumptions on V and Q, 6 > 0. Let

U = {x : \x - y\ < \6 for some y € V}. Then V C U and (/ C f2. Let x be the characteristic function of U, and choose a nonnegative G Cf (5{/ 2 (0)) such that f = 1. Then we can take / = x * 0 there exists xt £ K \ (J t Vj . Since K is compact, the xe have an accumulation point x £ K as e —• 0. But then i £ / { \ (Jj V} , which is absurd. I (0.19) Theorem. Let K C IRn be compact and let V\,..., Vjv be bounded open sets such that K C U f ^ • Then there exist functions Ci, • • •, Cn with Q € C f (V}) such that J2i Cj = 1 on KProof: Let W i , . . . , WN be as in Lemma (0.18). By Theorem (0.17), we can choose j G Cf(V}) with 0 < j < 1 and ^ = 1 on Wj. Then $ = J2i &J > 1 o n K, so we can take £,- = ^ ; - / $ , with the understanding that Cj = 0 wherever j = 0. I The collection of functions {^ J^ is called a partition of unity on K

subordinate to the covering {Vj}^.

14

Chapter 0

D. The Fourier Transform In this section we give a rapid introduction to the theory of the Fourier transform. For a more extensive discussion, see, e.g., Strichartz [47] or Folland [14], [17]. If / G Lx(IRn), its Fourier transform / is a bounded function on Mn defined by There is no universal agreement as to where to put the factors of 2TT in the definition of / , and we apologize if this definition is not the one the reader is used to. It has the advantage of making the Fourier transform both an isometry on I? and an algebra homomorphism from L1 (with convolution) to L°° (with pointwise multiplication). Clearly /(£) is well-defined for all £ and |j/|| T O < \\f\\i. Moreover: (0.20) Theorem. Gl 1 then (f*gT=fg. Proof: This is a simple application of Fubini's theorem:

(/ * gTXO = JJ e- 2 "" e /(* - y)ff(y) dy dx

= ho j c-2"»The Fourier transform interacts in a simple way with composition by translations and linear maps: (0.21) Proposition. Suppose f e L^W1). a. Iffa(x) = f(x+a) then ( / a f ( 0 = eMa^f(Ofa. If T is an invertible linear transformation of M", then (/ o T)~(£) = c. IfT is a rotation ofR", then (f o Tf = / o T. Proof: (a) and (b) are easily proved by making the substitutions y = x + a and y — Tx in the integrals defining (/a)~(£) and (/ o T)~*(0, respectively, (c) follows from (b) since T* — T~l and | det T\ — 1 when T is a rotation. I

Preliminaries

15

The easiest way to develop the other basic properties of the Fourier transform is to consider its restriction to the Schwartz class S. In what follows, if a is a multi-index, xaf denotes the function whose value at x is

x"f(x).

(0.22) Proposition. Suppose / £ §.

a. feC°° and ) -* (u, ) for every G Cc°°(n). Every locally integrable function u on Q can be regarded as a distribution by the formula {u, ) = f u, which accords with the notation introduced earlier. (The continuity follows from the Lebesgue dominated convergence theorem.) This correspondence is one-to-one if we regard two functions as the same if they are equal almost everywhere. Thus distributions can be regarded as "generalized functions." Indeed, we shall often pretend that distributions are functions and write (u, ) as $ u(x)(x) dx; this is a useful fiction that makes certain operations involving distributions more transparent. Every locally finite measure / i o n O defines a distribution by the formula (fj,,) = f dfi. In particular, if we take n to be the point mass at 0, we obtain the graddaddy of all distributions, the Dirac 5-function 6 G D ' defined by (6,) = (0). Theorem (0.13) implies that itu e L1, fu = a, and ue(x) = e~nu(e~lx), then ue —• a6 in T>' when e —> 0. If u, v € D'(fi), we say that u — v on an open set V C 0 if (u, ) — (t>, ) for all G C£°(V). The s u p p o r t of a distribution u is the complement of the largest open set on which u = 0. (To see that this is well-defined, one

Preliminaries

19

needs to know that if {V^}agyi is a collection of open sets and u — 0 on each Va, then « = 0 on (J Va. But if 6 C£°(|J Va), supp is covered by finitely many Va's. By means of a partition of unity on supp subordinate to this covering, one can write

= 1 on a neighborhood of supp u (by Theorem (0.17)). Then for any d> € Cc°° we have

This has two consequences. First, u is of "finite order": indeed, by (0.31) with K = JT,

Expanding da(ip4>) by the product rule, we see that

(0.32)

K«i,tf)i