An introduction to partial differential equations 2ed michael renardy robert c rogers ( TAM 13 2004 449s)

The second approach is to examine three partial differential equations Laplace's equation, the heat equation and the wave equation in a very elementary fashion again, this will probably

Texts in Applied Mathematics 13

Editors J.E Marsden

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Michael Renardy Robert C Rogers

Michael Renardy

Robert C Rogers

Department of Mathematics

460 McBlyde Hall

Virginia Polytechnic Institute

and State University

Conk01 and Dynamical Systems, 107-81 Division of Applied Mathematics

California Institute of Technology Brown University

Mathematics Subject Classification (2000): 35~01, 46~01, 47~01, 4 7 ~ 0 5

Library of Congress Cataloging~in~Publicatim Data

Renardy, Michael

An introduction to partial differential equations / Michael Renardy, Robert C Rogers.-

2nd ed

p cm - (Tents in applied mathematics ; 13)

Includes bibliographical references and index

ISBN 0~387~004440 (alk papey)

1 Differential equations, Parual I Rogers, Robert C I 1 Title 111 Series

QA374R4244 2003

ISBN 0~387~00444~0 Printed on acid~free paper

O 2004, 1993 SpringerVerlag New York, Inc

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Series Preface

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a bli~rring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied nlathematics This renewal of interest, both in r e search and teaching, has led to the establishnient of the serics Texts in Applied Matherrlatics (TAM)

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of excitement on the research Gontier as newer techniques, such as numeri- cal and symbolic conlputer systerns, dynamical systems, and chaos, mix with and reinforce the traditional ulethods of applied mathematics Thus, the purpose of this textbwk series is to meet the current and future needs

of these advances and to encourage the teaching of new courses

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J.E Marsden

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Preface

Partial differential equations are fundamental to the modeling of natural phenomena; they arise in every field of science Consequently, the desire to understand the solutions of these equations has always had a prominent place in the efforts of mathematicians; it has inspired such diverse fields as complex function theory, functional analysis and algebraic topology Like algebra, topology and rational mechanics, partial differential equations are

a core area of mathematics

Unfortunately, in the standard graduate curriculum, the subject is sel- dom taught with the same thoroughness as, say, algebra or integration theory The present book is aimed at rectifying this situation The goal of this course was to provide the background which is necessary to initiate work on a Ph.D thesis in PDEs The level of the book is aimed at be- ginning graduate students Prerequisites include a truly advanced calculus course and basic complex variables Lebesgue integration is needed only in Chapter 10, and the necessary tools from functional analysis are developed within the course

The book can be used to teach a variety of different courses Here at Vir- ginia Tech, we have used it to teach a four-semester sequence, but (more often) for shorter courses covering specific topics Students with some un- dergraduate exposure to PDEs can probably skip Chapter 1 Chapters 2-4 are essentially independent of the rest and can be omitted or postponed if the goal is to learn functional analytic methods as quickly as possible Only the basic definitions at the beginning of Chapter 2, the WeierstraD approxi- mation theorem and the Arzela-Ascoli theorem are necessary for subsequent

Chapter 12 uses some definitions from the beginning of Chapter 11) and can be covered in any order desired

We would like to thank the many friends and colleagues who gave us sug- gestions, advice and support In particular, we wish to thank Pave1 Bochev, Guowei Huang, Wei Huang, Addison Jump, Kyehong Kang, Michael Keane, Hong-Chul Kim, Mark Mundt and Ken Mulzet for their help Special thanks is due to Bill Hrusa, who read a good deal of the manuscript, some of

it with great care and made a number of helpful suggestions for corrections and improvements

Notes on the second edition

We would like to thank the many readers of the first edition who provided comments and criticism In writing the second edition we have, of course, taken the opportunity to make many corrections and small additions We have also made the following more substantial changes

each section with the easiest problems first

and elliptic systems

placed in a separate chapter Basic definitions, examples, and theo- rems appear at the beginning while technical lemmas are put off until the end New examples and problems have been added

"Young-measure" solutions

Contents

Series Preface

Preface

1 Introduction

1.1 Basic Mathematical Questions

1.1.1 Existence

1.1.2 Multiplicity

1.1.3 Stability

1.1.4 Linear Systems of ODES and Asymptotic Stability 1.1.5 Well-Posed Problems

1.1.6 Representations

1.1.7 Estimation

1.1.8 Smoothness

1.2 Elementary Partial Differential Equations

1.2.1 Laplace's Equation

1.2.2 The Heat Equation

1.2.3 The Wave Equation

2 Characteristics 2.1 Classification and Characteristics

2.1.1 The Symbol of a Differential Expression

2.1.2 Scalar Equations of Second Order

2.1.3 Higher-Order Equations and Systems

2.1.4 Nonlinear Equations

2.2 The Cauchy-Kovalevskaya Theorem

2.2.1 Real Analytic Functions

2.2.2 Majorization

2.2.3 Statement and Proof of the Theorem

2.2.4 Reduction of General Systems

2.2.5 A PDE without Solutions

2.3 Holmgren's Uniqueness Theorem

2.3.1 An Outline of the Main Idea

2.3.2 Statement and Proof of the Theorem

2.3.3 The WeierstraD Approximation Theorem

3 Conservation Laws and Shocks 3.1 Systems in One Space Dimension

3.2 Basic Definitions and Hypotheses

3.3 Blowup of Smooth Solutions

3.3.1 Single Conservation Laws

3.3.2 The p System

3.4 Weak Solutions

3.4.1 The Rankine-Hugoniot Condition

3.4.2 Multiplicity

3.4.3 The Lax Shock Condition

3.5 Riemann Problems

3.5.1 Single Equations

3.5.2 Systems

3.6 Other Selection Criteria

3.6.1 The Entropy Condition

3.6.2 Viscosity Solutions

3.6.3 Uniqueness

4 Maximum Principles 4.1 Maximum Principles of Elliptic Problems

4.1.1 The Weak Maximum Principle

4.1.2 The Strong Maximum Principle

4.1.3 A Priori Bounds

4.2 An Existence Proof for the Dirichlet Problem

4.2.1 The Dirichlet Problem on a Ball

4.2.2 Subharmonic Functions

4.2.3 The Arzela-Ascoli Theorem

4.2.4 Proof of Theorem 4.13

4.3 Radial Symmetry

4.3.1 Two Auxiliary Lemmas

4.3.2 Proof of the Theorem

4.4 Maximum Principles for Parabolic Equations

4.4.1 The Weak Maximum Principle

4.4.2 The Strong Maximum Principle 118

5 Distributions 5.1 Test Functions and Distributions

5.1.1 Motivation

5.1.2 Test Functions

5.1.3 Distributions

5.1.4 Localization and Regularization

5.1.5 Convergence of Distributions

5.1.6 Tempered Distributions

5.2 Derivatives and Integrals

5.2.1 Basic Definitions

5.2.2 Examples

5.2.3 Primitives and Ordinary Differential Equations 5.3 Convolutions and Fundamental Solutions

5.3.1 The Direct Product of Distributions

5.3.2 Convolution of Distributions

5.3.3 Fundamental Solutions

5.4 The Fourier Transform

5.4.1 Fourier Transforms of Test Functions

5.4.2 Fourier Transforms of Tempered Distributions 5.4.3 The Fundamental Solution for the Wave Equation 5.4.4 Fourier Transform of Convolutions

5.4.5 Laplace Transforms

5.5 Green's Functions

5.5.1 Boundary-Value Problems and their Adjoints 5.5.2 Green's Functions for Boundary-Value Problems 5.5.3 Boundary Integral Methods

6 Function Spaces 6.1 Banach Spaces and Hilbert Spaces

6.1.1 Banach Spaces

6.1.2 Examples of Banach Spaces

6.1.3 Hilbert Spaces

6.2 Bases in Hilbert Spaces

6.2.1 The Existence of a Basis

6.2.2 Fourier Series

6.2.3 Orthogonal Polynomials

6.3 Duality and Weak Convergence

6.3.1 Bounded Linear Mappings

6.3.2 Examples of Dual Spaces

6.3.3 The Hahn-Banach Theorem

6.3.4 The Uniform Boundedness Theorem

6.3.5 Weak Convergence

7 Sobolev Spaces 203

7.1 Basic Definitions 204

7.2 Characterizations of Sobolev Spaces 207

7.2.1 Some Comments on the Domain fl 207

7.2.2 Sobolev Spaces and Fourier Transform 208

7.2.3 The Sobolev Imbedding Theorem 209

7.2.4 Compactness Properties 210

7.2.5 The Trace Theorem 214

7.3 Negative Sobolev Spaces and Duality 218

7.4 TechnicalResults 220

7.4.1 Density Theorems 220

7.4.2 Coordinate Transformations and Sobolev Spaces on Manifolds 221

7.4.3 Extension Theorems 223

7.4.4 Problems 225

8 Operator Theory 228 8.1 Basic Definitions and Examples 229

8.1.1 Operators 229

8.1.2 Inverse Operators 230

8.1.3 Bounded Operators, Extensions 230

8.1.4 Examples of Operators 232

8.1.5 Closed Operators 237

8.2 The Open Mapping Theorem 241

8.3 Spectrum and Resolvent 244

8.3.1 The Spectra of Bounded Operators 246

8.4 Symmetry and Self-adjointness 251

8.4.1 The Adjoint Operator 251

8.4.2 The Hilbert Adjoint Operator 253

8.4.3 Adjoint Operators and Spectral Theory 256

8.4.4 Proof of the Bounded Inverse Theorem for Hilbert Spaces 257

8.5 Compact Operators 259

8.5.1 The Spectrum of a Compact Operator 265

8.6 Sturm-Liouville Boundary-Value Problems 271

8.7 The Fredholm Index 279

9 Linear Elliptic Equations 283

9.1 Defin~t~ons 283

9.2 Existence and Uniqueness of Solutions of the Dirichlet Problem 287

9.2.1 The Dirichlet Problem-Types of Solutions 287

9.2.2 The Lax-Milgram Lemma 290

9.2.3 Girding's Inequality 292

9.2.4 Existence of Weak Solutions 298

9.3 Eigenfunction Expansions

9.3.1 Fredholm Theory

9.3.2 Eigenfunction Expansions

9.4 General Linear Elliptic Problems

9.4.1 The Neumann Problem

9.4.2 The Complementing Condition for Elliptic Systems 9.4.3 The Adjoint Boundary-Value Problem

9.4.4 Agmon's Condition and Coercive Problems

9.5 Interior Regularity

9.5.1 Difference Quotients

9.5.2 Second-Order Scalar Equations

9.6 Boundary Regularity

10 Nonlinear Elliptic Equations 335 10.1 Perturbation Results 335

10.1.1 The Banach Contraction Principle and the Implicit Function Theorem 336

10.1.2 Applications to Elliptic PDEs 339

10.2 Nonlinear Variational Problems 342

10.2.1 Convex problems 342

10.2.2 Nonconvex Problems 355

10.3 Nonlinear Operator Theory Methods 359

10.3.1 Mappings on Finite-Dimensional Spaces 359

10.3.2 Monotone Mappings on Banach Spaces 363

10.3.3 Applications of Monotone Operators to Nonlinear PDEs 366

10.3.4 Nemytskii Operators 370

10.3.5 Pseudrrmonotone Operators 371

10.3.6 Application to PDEs 374

11 Energy Methods for Evolution Problems 11.1 Parabolic Equations

11.1.1 Banach Space Valued Functions and Distributions 11.1.2 Abstract Parabolic Initial-Value Problems

11.1.3 Applications

11.1.4 Regularity of Solutions

11.2 Hyperbolic Evolution Problems

11.2.1 Abstract Second-Order Evolution Problems

11.2.2 Existence of a Solution

11.2.3 Uniqueness of the Solution

11.2.4 Continuity of the Solution

12 Semigroup Methods 395 12.1 Semigroups and Infinitesimal Generators 397

12.1.1 Strongly Continuous Semigroups 397

xi" Contents

12.1.2 The Infinitesimal Generator

12.1.3 4 bstract ODES 12.2 The HilleYosida Theorem

12.2.1 The HilleYosida Theorem

12.2.2 The Lumer-Phillips Theorem

12.3 Applications to PDEs 12.3.1 Symmetric Hyperbolic Systems

12.3.2 The Wave Equation 12.3.3 The SchrGdinger Equation

12.4 Analytic Semigroups 12.4.1 4 nalytic Semigroups and Their Generators

12.4.2 Fractional Powers 12.4.3 Perturbations of Analytic Semigroups

12.4.4 Regularity of Mild Solutions

A References A.l Elementary Texts

A.2 Basic Graduate Texts

A.3 Specialized or Advanced Texts

A.4 Multivolume or Encyclopedic Works A.5 Other References

Introduction

This book is intended to introduce its readers to the mathematical theory

of partial differential equations But to suggest that there is a "theory"

of partial differential equations (in the same sense that there is a theory

of ordinary differential equations or a theory of functions of a single com- plex variable) is misleading PDEs is a much larger subject than the two mentioned above (it includes both of them as special cases) and a less well developed one However, although a casual observer may decide the subject

is simply a grab bag of unrelated techniques used to handle different types

of problems, there are in fact certain themes that run throughout

In order to illustrate these themes we take two approaches The first is

to pose a group of questions that arise in many problems in PDEs (ex- istence, multiplicity, etc.) As examples of different methods of attacking these problems, we examine some results from the theories of ODES, ad- vanced calculus and complex variables (with which the reader is assumed

to have some familiarity) The second approach is to examine three partial differential equations (Laplace's equation, the heat equation and the wave equation) in a very elementary fashion (again, this will probably be a re- view for most readers) We will see that even the most elementary methods foreshadow deeper results found in the later chapters of this book

2 1 Introduction

1.1 Basic Mathematical Questions

1.1.1 Existence

Questions of existence occur naturally throughout mathematics The ques-

tion of whether a solution exists should pop into a mathematician's head

any time he or she writes an equation down Appropriately, the problem of existence of solutions of partial differential equations occupies a large por- tion of this text In this section we consider precursors of the PDE theorems

to come

Initial-value problems in ODEs

The prototype existence result in differential equations is for initial-value problems in ODEs

be an open set, and let F : D + Rn be continuous i n its first variable and uniformly Lipschitz i n its second; i.e., for (t, y) t D, F ( t , y) is

continuous as a function o f t , and there exists a constant y such that for any (t, y l ) and (t, yq) i n D we have

Then, for any (to, yo) t D , there exists an interval I := ( t , t+) containing

to, and at least one solution y t C1(I) of the initial-value p r o b l e m

The proof of this can be found in almost any text on ODEs We make note of one version of the proof that is the source of many techniques in PDEs: the construction of an equivalent integral equation In this proof, one shows that there is a continuous function y that satisfies

Then the fundamental theorem of calculus implies that y is differentiable and satisfies (1.2), (1.3) (cf the results on smoothness below) The solution

of (1.4) is obtained from an iterative procedure; i.e., we begin with an initial guess for the solution (usually the constant function yo) and proceed to

1.1 Basic Mathematical Questions 3

Existence theorems of advanced calculus

The following theorems from advanced calculus give information on the solution of algebraic equations The first, the inverse function theorem, considers the problem of n equations in n unknowns

T h e o r e m 1.2 ( I n v e r s e f u n c t i o n t h e o r e m ) Suppose the function

is nonsingular Then there is a neighborhood N, o f x o and a neighborhood

Np of PO such that F : N, + Np is one-to-one and onto; i.e., for evellj

p t Np the equation

has a unique solution i n N,

Our second result, the implicit function theorem, concerns solving a

T h e o r e m 1.3 ( I m p l i c i t f u n c t i o n t h e o r e m ) Suppose the function

F : Rq x Rp 3 ( x , y ) ti F ( x , y ) t Rp

is C 1 i n a neighborhood of a point (xo, yo) Further assume that

F ( X O , Y O ) = 0,

4 1 Introduction

and that the p x p matrix

is nonsingular Then there is a neighborhood N, c IWq o f x o and a function

y : N, + IWp such that

and for every x t N,

The two theorems illustrate the idea that a nonlinear system of equations behaves essentially like its linearization as long as the linear terms dominate the nonlinear ones Results of this nature are of considerable importance

in differential equations

1.1.2 Multiplicity

Once we have asked the question of whether a solution to a given problem exists, it is natural to consider the question of how many solutions there are

Uniqueness for initial-value problems in ODEs

The prototype for uniqueness results is for initial-value problems in ODEs

Theorem 1.4 (ODE uniqueness) Let the function F satisfy the hypo- theses of Theorem 1.1 Then the initial-value problem (1.2), (1.3) has at most one solution

It should be noted that although this result covers a very wide range of initial-value problems, there are some standard, simple examples for which uniqueness fails For instance, the problem

has an entire family of solutions parameterized by y t [O, 11:

1.1 Basic Mathematical Questions 5

Nonuniqueness for linear and nonlinear boundary-value problems

While uniqueness is often a desirable property for a solution of a problem (often for physical reasons), there are situations in which multiple solutions

tions is an eigenvalue problem The reader should, of course, be familiar with the various existence and multiplicity results from finite-dimensional linear algebra, but let us consider a few problems from ordinary differential equations We consider the following second-order ODE depending on the

Of course, if we imposed two initial conditions (at one point in space) Theorem 1.4 would imply that we would have a unique solution (To apply the theorem directly we need to convert the problem from a second-order equation to a first-order system.) However, if we impose the two-point boundary conditions

the uniqueness theorem does not apply Instead we get the following result

Theorem 1.5 There are two alternatives for the solutions of the boundary-value problem (1.61, (1.71, (1.8)

1 IfX = A, := ((2n+1)27r2)/4, n = 0,1,2, , then the boundary-value problem has a family of solutions parameterized by A t ( - a , a ) :

u , (x) = A sin (2n + 1 ) ~

In this case we say X is an eigenvalue

2 For all other values of X the only solution of the boundary-value problem is the trivial solution

This characteristic of having either a unique (trivial) solution or an infi- nite linear family of solutions is typical of linear problems More interesting multiplicity results are available for nonlinear problems and are the main

subject of modern bifurcation theory For example, consider the following

nonlinear boundary-value problem, which was derived by Euler to describe the deflection of a thin, uniform, inextensible, vertical, elastic beam under

a load A:

Figure 1.1 Bifurcation diagram for the nonlinear boundary-value problem

(Note that the linear ODE (1.6) is an approximation of (1.9) for small 8.) Solutions of this nonlinear boundary-value problem have been computed

in closed form (in terms of Jacobi elliptic functions) and are probably best

displayed by a bifurcation diagram such as Figure 1.1 This figure displays

for every A) Note that a branch of nontrivial solutions emanates from each

1 , 2 , 3 , , there are precisely 2n nontrivial solutions of the boundary-value problem

1.1.3 Stability

The term stability is one that has a variety of different meanings within mathematics One often says that a problem is stable if it is "continuous with respect to the data"; i.e., a problem is stable if when we change the problem "slightly," the solution changes only slightly We make this precise below in the context of initial-value problems for ODEs Another notion of stability is that of "asymptotic stability." Here we say a problem is stable if all of its solutions get close to some "nice" solution as time goes to infinity

We make this notion precise with a result on linear systems of ODEs with constant coefficients

Stability with respect to initial conditions

and we define y (t, to, yo) to be the unique solution of (1.2), (1.3) We then have the following standard result

1.1 Basic Mathematical Questions 7

Theorem 1.6 (Continuity with respect to initial conditions) The function y is well defined on an open set

I (to, YO) - (Zo, Y o ) < 6,

then y(t,&, 40) is well defined and

Thus, we see that small changes in the initial conditions result in small changes in the solutions of the initid-value problem

1.1.4 Linear Systems of ODEs and Asymptotic Stability

We now examine a concept called asymptotic stability in the context of linear system of ODEs We consider the problem of finding a function

y : R + Rn that satisfies

dy(t) = A(t)y(t)+f(t),

where to t R, yo t Rn, the vector valued function f : R + Rn and the

Asymptotic stability describes the behavior of solutions of homogeneous systems as t goes to infinity

Definition 1.7 The linear homogeneous system

1 asymptotically stable if every solution of (1.15) satisfies

lim 1 y(t)l = 0,

2 completely unstable if every nonzero solution of (1.15) satisfies

lim y(t)l = oo

The following fundamental result applies to constant coefficient systems

Theorem 1.8 Let A t R n X n be a constant matrix with eigenvalues

A1, A 2 , , A n Then the linear homogeneous system of ODES

1 asymptotically stable if and only if all the eigenvalues of A have negative real parts; and

2 completely unstable if and only if all the eigenvalues o f A have positive real parts

The proof of this theorem is based on a diagonalization procedure for

problem associated with (1.18)

Formula 1.19 is the precursor of formulas in semigroup theory that we encounter in Chapter 12

1.1.5 W e l l - P o s e d Problems

We say that a problem is well-posed (in the sense of Hadamard) if

1 there exists a solution,

2 the solution is unique

If these conditions do not hold, a problem is said to be ill-posed Of course, the meaning of the term continuity with respect to the data has to be made

more precise by a choice of norms in the context of each problem considered

In the course of this book we classify most of the problems we encounter

as either well-posed or ill-posed, but the reader should avoid the assumption that well-posed problems are always "better" or more "physically realistic" than ill-posed problems As we saw in the problem of buckling of a beam mentioned above, there are times when the conditions of a well-posed prob- lem (uniqueness in this case) are physically unrealistic The importance of ill-posedness in nature was stressed long ago by Maxwell [Max]:

1.1 Basic Mathematical Questions 9

For example, the rock loosed by frost and balanced on a sin- gular point of the mountain-side, the little spark which kindles the great forest, the little word which sets the world afighting, the little scruple which prevents a man from doing his will, the little spore which blights all the potatoes, the little gemmule which makes us philosophers or idiots Every existence above

a certain rank has its singular points: the higher the rank, the more of them At these points, influences whose physical mag- nitude is too small to be taken account of by a finite being may produce results of the greatest importance All great results produced by human endeavour depend on taking advantage of these singular states when they occur

We draw attention to the fact that this statement was made a full century before people "discovered" all the marvelous things that can be done with cubic surfaces in R3

1.1.6 Representations

There is one way of proving existence of a solution to a problem that is more satisfactory than all others: writing the solution explicitly In addition to the aesthetic advantages provided by a representation for a solution there are many practical advantages One can compute, graph, observe, estimate, manipulate and modify the solution by using the formula We examine below some representations for solutions that are often useful in the study

of PDEs

Variation of parameters

Variation of parameters is a formula giving the solution of a nonho-

some utility in terms of actually computing solutions, its primary use is analytical

The key to the variations of constants formula is the construction of a

fundamental solution matrix + ( t , r ) t R n X n for the linear homogeneous system This solution matrix satisfies

mental matrix is standard and is left as an exercise Note that the unique

system is given by

~ ( t ) := + ( t , t o ) ~ o (1.23)

The use of Leibniz' formula reveals that the variation of parameters formula

gives the solution of the initial-value problem ( 1 3 ) (1.14) for the

nonhomogeneous system

Cauchy's integral formula

Cauchy's integral formula is the most important result in the theory of complex variables It provides a representation for analytic functions in terms of its values at distant points Note that this representation is rarely used to actually compute the values of an analytic function; rather it is used to deduce a variety of theoretical results

Theorem 1.9 (Cauchy's integral formula) Let f be analytic i n a

simply connected domain D c C and let C be a simple closed positively oriented curve i n D Then for any point zo i n the interior of C

1 l 7 Estimation

When we speak of an estimate for a solution we refer to a relation that

gives an indication of the solution's size or character Most often these are inequalities involving norms of the solution We distinguish between

the following two types of estimate An a posteriori estimate depends on

knowledge of the existence of a solution This knowledge is usually obtained

through some sort of construction or explicit representation An a priori

estimate is one that is conditional on the existence of the solution; i.e., a result of the form, "If a solution of the problem exists, then it satisfies "

We present here an example of each type of estimate

Gronwall's inequality and energy estimates

In this section we derive an a priori estimate for solutions of ODES that is

related to the energy estimates for PDEs that we examine in later chapters

The uniqueness theorem 1.4 is an immediate consequence of this result To

derive our estimate we need a fundamental inequality called Gronwall's inequality

Lemma 1.10 (Gronwall's inequality) Let

u : [a,bl + [ o , ~ ) ,

u : [a, b] + R,

1.1 Basic Mathematical Questions 11

be continuous functions and let C be a constant Then if

for t t [a, b], it follows that

f o r t t [a,b]

The proof of this is left as an exercise

Lemma 1.11 (Energy estimate for ODEs) Let F : R x Rn + Rn

satisfy the hypotheses of Theorem 1.1, i n particular let it be uniformly Lip- schitz i n its second variable with Lipschitz constant y (cf (1.1)) Let yl

and y2 be solutions of (1.2) on the interval [to, T I ; i.e.,

~ : ( t ) = F ( t , ~ i ( t ) )

for i = 1,2 and t t [to, T I Then

Yl(t) - Yz(t)I2 < Yl(to) - Y Z ( ~ O ) ~ 2e2y(t-to), (1.28)

Proof We begin by using the differential equation, the Cauchy-Schwarz inequality and the Lipschitz condition to derive the following inequality

Now (1.28) follows directly from Gronwall's inequality

results are indeed obtained a priori: nothing we did depended on the exis-

tence of a solution, only on the equations that a solution would satisfy if it

did exist

Maximum principle for analytic functions

As an example of an a posteriori result we consider the following theorem

Theorem 1.12 (Maximum modulus principle) Let D c C be a bounded domain and let f be analytic on D and continuous on the closure

of D Then 1 f 1 achieves its maximum on the boundary of D ; i.e., there exists zo t a D such that

The reader is encouraged to prove this using Cauchy's integral formula (cf Problem 1.10) Such a proof, based on an explicit representation for the

function f , is a posteriori We note, however, that it is possible to give an

a prior2 proof of the result; and Chapter 4 is dedicated to finding a priori

maximum principles for PDEs

1.1.8 Smoothness

One of the most important modern techniques for proving the existence of

a solution to a partial differential equation is the following process

1 Convert the original PDE into a "weak" form that might conceivably have very rough solutions

2 Show that the weak problem has a solution

3 Show that the solution of the weak equation actually has more smoothness than one would have at first expected

4 Show that a "smooth solution of the weak problem is a solution of the original problem

We give a preview of parts one, two, and four of this process in Section 1.2.1 below, but in this section let us consider precursors of the methods for part three: showing smoothness

Smoothness of solutions of ODES

The following is an example of a "bootstrap" proof of regularity in which

can be used to prove the regularity portion of Theorem 1.1 (which asserted the existence of a C1 solution)

Theorem 1.13 If F : R x Rn + Rn is i n Cm-'(R x Rn) for some integer

m > 1, and y t CO(R) satisfies the integral equation

t

Y = Y O+ / F ( s , Y ( 3 ) ) ds, (1.30)

t o then i n fact y t Cm(R)

1.1 Basic Mathematical Questions 13

Proof Since F(s, y ( s ) ) is continuous, we can use the Fundamental Theorem

differentiable, so the left-hand side must be as well, and

~ ' ( t ) = F ( t , ~ ( t ) ) (1.31)

Thus, y t C1(R) If F is in C1, we can repeat this process by noting that the right-hand side of (1.31) is differentiable (so the left-hand side is as well) and

so y t C2(R) This can be repeated as long as we can take further con-

Smoothness of analytic functions

integral formula

Theorem 1.14 If a function f : C + C is analytic at zo t C (i.e., if

it has at least one complex derivative i n a neighborhood o f z o ) , then it has complex derivatives of arbitrary order In fact,

for any simple, closed, positively oriented curve C lying i n a simply connected domain i n which f is analytic and having zo i n its interior

The proof can be obtained by differentiating Cauchy's integral formula

(1.25) under the integral sign This is a common technique in PDEs, and

Problems

1.1 Let yi, be the sequence defined by (1.5) Show that

l ~ k + l ( t ) - ~ic(t)l < Y (t - to) max Iyk(.r) - yk-l(.r)I

r t [to ,tl Use this to show that the sequence converges uniformly for to < t < T for any T < to + 117

1.2 Use the implicit function theorem to determine when the equation

z2 + y2 + z2 = 1

result

1.3 Show that if F as described in Theorem 1.3 is Ck, then y is Ck as

1.4 Show that there is an infinite family of minimizers of

1 ( u ) := u(tl2 + (1 - ( ~ ~ ( t ) ) ~ ) ~ dt satisfying u(0) = u(1) = 0 Hint: Use a sequence of the solutions of the

0 Remark: Minimization problems with features like these arise in the modeling of phase transitions

1.6 Give an example that shows that 6 in Theorem 1.6 cannot be chosen

1.7 Prove Theorem 1.8 in the case where the eigenvalues of A are distinct

1.8 Prove the existence and uniqueness of the solution of (1.21), (1.22)

1.9 Prove Gronwall's inequality

1.10 Prove Theorem 1.12 using Cauchy's integral formula

1 2 Elementary Partial Differential Equations

In the last section we discussed the basic types of mathematical questions that are considered throughout the rest of this book, and we looked at how those questions had been answered by two subdisciplines of PDEs: ODES and complex variables We now look at how these questions are often approached in elementary courses on partial differential equations To do this, we consider three basic PDEs (Laplace's equation, the heat equation and the wave equation) Although we sometimes use an analytical approach

to investigate their character, our basic technique is the explicit calculation

of solutions At this point we are not terribly concerned with either rigor

or generality but rather with foreshadowing material to come; all of the methods and observations presented here are generalized later on

1.2 Elementary Partial Differential Equations 15 1.2.1 Laplace's Equation

Perhaps the most important of all partial differential equations is

known as Laplace's equation You will find applications of it to problems in gravitation, elastic membranes, electrostatics, fluid flow, steady-state heat conduction and many other topics in both pure and applied mathematics

As the remarks of the last section on ODES indicated, the choice of boundary conditions is of paramount importance in determining the well- posedness of a given problem The following two common types of boundary

will be studied in a more general context in later chapters

Dirichlet conditions Given a function f : an + R, we require

In the context of elasticity, u denotes a change of position, so Dirichlet

boundary conditions are often referred to as displacement conditions

Neumann conditions Given a function f : an + R, we require

as a force, so Neumann boundary conditions are often referred to as traction

boundary conditions

and f , and the function space in which we wish u to lie These are central

areas of concern in later chapters

Solution by separation of variables

The first method we present for solving Laplace's equation is the most widely used technique for solving partial differential equations: separation

of variables The technique involves reducing a partial differential equation

to a system of ordinary differential equations and expressing the solution

of the PDE as a sum or infinite series

Let us consider the following Dirichlet problem on a square in the plane Let

at each point in fl and satisfying the boundary conditions

When we put a function of this form into (1.36), the partial derivatives in

and Y; i.e., (1.36) becomes

At any point (2, y) at which u is nonzero we can divide this equation by

u and rearrange to get

We now argue as follows: Since the right side of the equation does not depend on the variable x, neither can the left side; likewise, since the left side does not depend on y, neither does the right side The only function

on the plane that is independent of both x and y is a constant, so we must have

This gives us

Solving these equations and using (1.41), we get the following four- parameter family of solutions of the differential equation (1.36):

(Since we can verify directly that each of these functions is indeed a solution

of the differential equation (1.36), there is no need to make the formal argument used to derive (1.45) and (1.46) rigorous.)

The more interesting aspect of separation of variables involves finding a combination of the solutions in (1.47) that satisfies given boundary con- ditions (and justifying this combination rigorously) In the rather simple set of boundary conditions chosen above, enforcing the three conditions

1.2 Elementary Partial Differential Equations 17

(1.37), (1.38) and (1.39) reduces the family (1.47) to the following infinite collection

is rigged to be a finite linear combination of sine functions,

then we can simply take

and define

Since this is a finite sum, we can differentiate term by term; so u satisfies the differential equation (1.36) The boundary conditions can be confirmed simply by plugging in the boundary points

However, the question remains: What is to be done about more general

on heat conduction Fourier claimed, in effect, that "any" function f could

be "represented by an infinite trigonometric series (now referred to as a Fourier sine series):

m

n=1 The removal of the quotation marks from the sentence above was one of the more important mathematical projects of the nineteenth century Specif- ically, mathematicians needed to specify the type of convergence implied

by the representation (1.52) and then identify the class of functions that can be achieved by that type of convergence.' In later chapters we describe some of the main results in this area, but for the moment let us just accept Fourier's assertion and try to deduce its consequences

The first question we need to consider is the determination of the Fourier coefficients or, The key here is the mutual orthogonality of the sequence

'Anyone interested in the history of mathematics or the philosophy of science will find the history of Fomier's work fascinating In the early nineteenth century the entire notion

of convergence and the meaning of infiwte series was not well formulated Lagrange and his cohorts in the Academy of Sciences in Paris criticized Fomier for his lack of rigor Although they were technically correct, they were essentially castigating Fomier for not having produced a body of mathematics that it took generations of mathematicians (including the likes of Cauchy) to fiwsh

of sine functions making up our series That is,

Here 6ij is the Kronecker delta:

Thus, if we proceed formally and multiply (1.52) by sin j n x and integrate,

where

It remains to answer the following questions:

r Is the limit of the series differentiable, and if so, does it satisfy (1.36)? That is, can we take the derivatives under the summation sign?

problem? More generally, is the problem well-posed?

All of these questions will be answered in a more general context in later chapters

Example 1.15 Let us ignore for the moment the theoretical questions that remain to be answered and do a calculation for a specific problem We wish to solve the Dirichlet problem (1.36)-(1.40) with data

1.2 Elementary Partial Differential Equations 19

We begin by calculating the Fourier coeficients o f f using (1.55);

Note that the even coefficients vanish Thus, we can modify (1.56) to get the following separation of variables solution of our Dirichlet problem

Poisson's integral formula in the upper half-plane

In this section we describe Poisson's integral formula i n the upper half-

plane This formula gives the solution of Dirichlet's problem in the upper half-plane It is often derived in elementary complex variables courses

satisfies Laplace's equation (1.36) in the upper half-plane and and that it

boundary conditions

for x t R

Poisson's integral formula is an example of the use of integral operators

to solve boundary-value problems In later chapters we will generalize the

technique through the use of Green's functions

Variational formulations

In this section we give a demonstration of a variational technique for prov-

ing the existence of solutions of Dirichlet's problem on a "general" domain

(since as we shall see in later chapters its rigorous application requires some rather heavy machinery), but it is presented in many elementary courses (particularly in Physics and Engineering) using the formal arguments we sketch here

We begin by defining an energy functional

and a class of admissible functions

A := {U : n + R 1 U(X) = f (x) for x t an, E(U) i a ) (1.64)

We can now show the following

Theorem 1.16 I f A is nonempty, and ifthere exists a t A that minimizes

E over A; i.e.,

then a is a solution of the Dirichlet problem

Before giving the proof we note that there are some serious questions to

be answered before this theorem can be applied

to satisfy so that f can be extended into n using a function of finite energy?

2 Does there exist a minimizer a t A ?

These questions are often ignored (either explicitly or tacitly) in elementary presentations, but we shall see that they are far from easy to answer

Proof We give only a sketch of the proof and that will contain a number

of holes to be filled later on Let us define

A := {U : n + R 1 U(X) = o for x t an, E(u) i a} (1.66)

u t Ao, then (u + tu) t A for any t t R We take any u t Ao and define a function or : R + R by

~ ( t ) := & ( a + tu)

= L { V a l 2 + 2 t V ~ V u + t 2 V u 2 } dx (1.67)

at t = 0 yields

equation is aversion of the fundamental lemma of the calculus ofvariations

(This name has been given to a wide range of results that allow one to deduce that a function that satisfies a variational equation such as (1.68) also satisfies a pointwise differential equation Another name commonly used in the same way is the DuBois-Reymond lemma in honor of the first versions of such a result.) We now prove a very weak version of this result

1.2 Elementary Partial Differential Equations 21

Lemma 1.17 Let F : n + Rn be i n C1(n) and satisfy the variational equation

for evellj u t A with compact support Then

Since div F is continuous, if there is a point xo at which it is nonzero (without loss of generality let us assume it is positive there) there is a ball

B around xo contained in n such that div F > 6 > 0 We can then use a function 0 whose graph is a positive "blip" inside of B and zero outside of

B (such a function is easy to construct, and the task is left to the reader)

to obtain

This is a contradiction, and the proof of the lemma is complete

Now to complete the proof of the theorem, we note that if u is in C 2 ( n )

we can use Lemma 1.17 and (1.68) to deduce

about its smoothness Thus, the completion of this proof awaits the results

on elliptic regularity of later chapters

Equation (1.68) is known as the weak form of Laplace's equation We refer to a solution of (1.33) as a strong solution and a solution of (1.68) as

a weak solution of Laplace's equation We will generalize these notions to

many other types of equations in later chapters

Note that every strong solution of Laplace's equation is also a weak solution To see this, we simply multiply (1.33) by an arbitrary function

22 1 Introduction

u = 0 on ail This gives

o=L ( A ~ ) ~ ~ ~ = - ~ v ~ V ~ ~i ~V u V u d x + /(1.75) _ ~ V ~ ~ ~ S =

-However, as we noted above when we showed that a solution of the min-

imum energy problem was a weak solution of Laplace's equation, unless

we know more about the continuity of a weak solution we cannot show

it is a strong solution This is a common theme in the modern theory of

PDEs It is often easy to find some sort of weak solution to an equation, but

relatively hard to show that the weak solution is in fact a strong solution

Problems

1.12 Compute the Fourier sine series coefficients for the following func-

tions defined on the interval [O, 11

4

1.13 Write a computer program that calculates partial sums of the series

defined above and displays them graphically superimposed on the limiting

function

1.14 A function on the interval [O, 11 can also be expanded in a Fourier

cosine series of the form

Derive a formula for the cosine coefficients

1.15 Compute the Fourier cosine coefficients for the functions given in

Problem 1.12 Use a modification of the computer program developed in

Problem 1.13 to display partial sums of the cosine series

1.16 Both the Fourier sine and cosine series given above converge not

only in the interval [O, 11, but on the entire real line If one computed both

the sine and cosine series for the functions graphed below, what would you

expect the respective graphs of the limits of the series to be on the whole

real line

1.2 Elementary Partial Differential Equations 23

1.17 Solve Laplace's equation on the square [O, 11 x [O, 11 for the following boundary conditions:

1.18 Verify that the Laplacian takes the following form in polar coordinates in R2:

1.19 Use the method of separation of variables t o find solutions of Laplace's equation of the form

1.20 Use the divergence theorem t o derive Green's identity

24 1 Introduction

1 2 2 The Heat Equation

The next elementary problem we examine is the heat equation:

Here u is a real-valued function depending on "spatial" variables x t Rn and on "time" t t R, and the operator A is the Laplacian defined in (1.33) which is assumed to act only on the spatial variables (XI, , x,) (The reason for the quotation marks above is that in the next section we will describe the "type" of a differential equation in a way that is independent

of any particular interpretation of the independent variables as spatial or temporal However, even after we have done this, we will often lapse back

to the terminology of space and time in order to draw analogies to the elementary Laplace, wave and heat equation described in this chapter.) As the name suggests, (1.77) describes the conduction of heat (with the de-

it governs a range of physical phenomena described as diffusive

In discussing typical boundary conditions we confine ourselves to prob- lems posed on a cylinder in space-time: fl? := {(x, t) t fl x (0, a ) } where

we place one initial condition on the solution We let 8 : fl + R be a given function and require

for x t afl and t t (0, a ) In problems of heat conduction, this corresponds

to placing a portion of the boundary in contact with a constant temperature source (an ice bath, etc.) Of course, such conditions can be identified with Dirichlet conditions for Laplace's equation

H e a t flux conditions Here we fix the normal derivative of u on some portion of the boundary

for x t afl and t t ( O , a ) , where n is the unit outward normal to afl

A simplified version of Fourier's law of heat conduction says that the heat

flux vector q at a point x at time t is given by

tion (1.80) can be thought of as fixing the flow of heat through a portion of

1.2 Elementary Partial Differential Equations 25

Laplace's equation should be obvious

Linear radiation conditions Here we require

for x t an and t t (O,cm), where or is a positive constant This can be

thought of as the linearization of Stefan's radiation law

about a steady-state solution of the boundary-value problem Stefan's law describes the loss of heat energy of a body through radiation into its surroundings

Solution by separation of variables

As part of our review of elementary solution methods we now examine the solution of a one-dimensional heat conduction problem by the method of

separation of variables We consider the following initial/boundary-value

Again, we make the argument that since the right side of the equation is independent o f t and the left side is independent of x , each side must be