Elliptic & Parabolic Equations

Elliptic & Parabolic Equations Zhuoqun Wu, Jingxue Yin & Chunpeng Wang

Elliptic & Parabolic Equations

Elliptic & Parabolic Equations Zhuoqun Wu, Jingxue Yin & Chunpeng Wang Jilin University, China

\JJS World Scientific N E W JERSEY

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ELLIPTIC AND PARABOLIC EQUATIONS Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-270-025-0 ISBN 981-270-026-9 (pbk)

Printed in Singapore by B & JO Enterprise

Preface

Elliptic equations and parabolic equations are two important branches in the field of partial differential equations. These two kinds of equations arise frequently at the same time in many applications: elliptic equations arise from the stationary case and parabolic equations from the nonstationary case. In history, theories of them have been developed almost simultaneously. Many methods applied to elliptic equations are also available to parabolic equations, although some new methods are required to be developed for the latter. So far there have been numerous monographs focusing separately on each kind of equations, see [Ladyzenskaja and Ural'ceva (1968)], [Ladyzenskaja, Solonnikov and Ural'ceva (1968)], [Gilbarg and Trudinger (1977)], [Lieberman (1996)], [Chen and Wu (1997)], [Chen (2003)] and [Gu (1995)]. However, there are very few books treating them in combination. In this respect, the book [Oleinik and Radkevic (1973)] should be mentioned, in which the equations considered include not only both linear elliptic and parabolic equations, but also all kinds of linear degenerate elliptic equations of second order. However, in the framework of this book, parabolic equations are regarded as degenerate elliptic equations by treating the time variable and space variables equally and thus only the commonalities between these two kinds of equations are presented. As a matter of course, in such a book, it is impossible to discuss deeply the specific properties of each kind. Prom our own experiences of teaching and research, we are aware of the necessity of writing a book which merges these two kinds of equations into an organic whole, involving the related basic theories and methods. This book is the result of a try following this idea, which is completed on the basis of lectures for graduate students majored in partial differential equations at Jilin University of China. The lectures have also been used at

VI

Elliptic and Parabolic

Equations

the summer school for graduate students in China. The purpose of this book is to provide an introduction to elliptic and parabolic equations of second order for graduate students and young scholars who want to work in the field of partial differential equations. It is our hope that the book will be beneficial not only to stress the commonalities between these two kinds of equations, but also to expose the specific properties of each kind, so that the readers can efficiently learn the related knowledge by observing the relationship and contrasting the similarities and differences. An exhaustive theory of these two kinds of equations is outside the scope of this book. The book covers only the related basic theories and methods in a reasonable volume. More attention is paid to typical equations. In treating each kind of equations, usually we first give a detailed discussion on some typical equations and then discuss general equations in a brief fashion. Our principal intention is to prevent the complicate derivation due to the generality of equations in form from concealing and obscuring substantial of the argument. Emphasis is put on introducing methods and techniques rather than collecting theorems and facts. The book consists of thirteen chapters. Some preliminary knowledge needed in the book is collected in Chapter 1, the main part of which is an introduction to the theory of Sobolev spaces and Holder spaces. Linear equations are discussed in Chapter 2 through Chapter 9. Chapter 2 and Chapter 3 are devoted to weak solutions and the L2 theory of linear elliptic equations and parabolic equations respectively. Properties of weak solutions are discussed in Chapter 4 and Chapter 5. In Chapter 4, we introduce two important methods, the De Giorgi iteration and the Moser iteration which are described only for some typical equations and applied only to the maximum estimates on weak solutions. Chapter 5 discusses Harnack's inequalities. In Chapter 6 and Chapter 7, we establish Schauder's estimates for elliptic equations and parabolic equations respectively. Based on these estimates, we prove the existence of classical solutions in Chapter 8. In establishing Schauder's estimates, we apply Campanato's approach which is based on the important fact that the Holder continuity of functions can be described in an equivalent integral form. By means of this approach, the proof is relatively simple. Chapter 9 is an argument of the Lp estimates which are used to discuss the existence of strong solutions. The solvability of quasilinear equations is studied in Chapter 10 through

Preface

vn

Chapter 12. Three methods are introduced, they are: the fixed point method (Chapter 10), the topology degree method (Chapter 11) and the monotone method (Chapter 12). The book finishes with Chapter 13, which contains an investigation of elliptic and parabolic equations with degeneracy. The first part of this chapter deals with linear equations, namely, equations with nonnegative characteristic form. Quasilinear equations are discussed in the second part of this chapter. As space is limited, we are not able to cover the study of fully nonlinear elliptic and parabolic equations in the book.

Wu Zhuoqun Yin Jingxue Wang Chunpeng

Jilin University, P. R. China August, 2006

Contents

Preface

v

1.

1

Preliminary Knowledge 1.1 Some Frequently Applied Inequalities and Basic Techniques 1.1.1 Some frequently applied inequalities 1.1.2 Spaces Ck(Q) and C#(0) 1.1.3 Smoothing operators 1.1.4 Cut-off functions 1.1.5 Partition of unity 1.1.6 Local flatting of the boundary 1.2 Holder Spaces 1.2.1 Spaces Ck'a(ty and Ck>a(n) 1.2.2 Interpolation inequalities 1.2.3 Spaces C2k+a>k+a/2(QT) 1.3 Isotropic Sobolev Spaces 1.3.1 Weak derivatives 1.3.2 Sobolev spaces Wk'p(Q) and W0fe'p(ft) 1.3.3 Operation rules of weak derivatives 1.3.4 Interpolation inequality 1.3.5 Embedding theorem 1.3.6 Poincare's inequality 1.4 t-Anisotropic Sobolev Spaces

1 1 2 3 5 6 6 7 7 8 13 14 14 15 17 17 19 21 24

Spaces W£kk(QT), Wf'k(QT), and V(QT) Embedding theorem Poincare's inequality

24 26 28

1.4.1 1.4.2 1.4.3

ix

V2(QT)

x

Elliptic and Parabolic

1.5

2.

3.

4.

Trace 1.5.1 1.5.2 1.5.3

Equations

of Functions in i/ x (fi) Some propositions on functions in H1(Q+) Trace of functions in tf^fi) Trace of functions in ^ ( Q r ) = W^21,1(Qr)

29 29 33 35

L 2 Theory of Linear Elliptic Equations

39

2.1 Weak 2.1.1 2.1.2 2.1.3 2.1.4

Solutions of Poisson's Equation Definition of weak solutions Riesz's representation theorem and its application . . Transformation of the problem Existence of minimizers of the corresponding functional 2.2 Regularity of Weak Solutions of Poisson's Equation . . . . 2.2.1 Difference operators 2.2.2 Interior regularity 2.2.3 Regularity near the boundary 2.2.4 Global regularity 2.2.5 Study of regularity by means of smoothing operators 2.3 L2 Theory of General Elliptic Equations 2.3.1 Weak solutions 2.3.2 Riesz's representation theorem and its application . . 2.3.3 Variational method 2.3.4 Lax-Milgram's theorem and its application 2.3.5 Fredholm's alternative theorem and its application .

39 40 41 43

L2 Theory of Linear Parabolic Equations

71

3.1 Energy Method 3.1.1 Definition of weak solutions 3.1.2 A modified Lax-Milgram's theorem 3.1.3 Existence and uniqueness of the weak solution . . . . 3.2 Rothe's Method 3.3 Galerkin's Method 3.4 Regularity of Weak Solutions 3.5 L2 Theory of General Parabolic Equations 3.5.1 Energy method 3.5.2 Rothe's method 3.5.3 Galerkin's method

71 72 73 75 79 85 89 94 94 96 97

De Giorgi Iteration and Moser Iteration

44 47 47 50 53 56 58 60 60 61 62 64 67

105

Contents

4.1 Global Boundedness Estimates of Weak Solutions of Poisson's Equation 4.1.1 Weak maximum principle for solutions of Laplace's equation 4.1.2 Weak maximum principle for solutions of Poisson's equation 4.2 Global Boundedness Estimates for Weak Solutions of the Heat Equation 4.2.1 Weak maximum principle for solutions of the homogeneous heat equation 4.2.2 Weak maximum principle for solutions of the nonhomogeneous heat equation 4.3 Local Boundedness Estimates for Weak Solutions of Poisson's Equation 4.3.1 Weak subsolutions (supersolutions) 4.3.2 Local boundedness estimate for weak solutions of Laplace's equation 4.3.3 Local boundedness estimate for solutions of Poisson's equation 4.3.4 Estimate near the boundary for weak solutions of Poisson's equation 4.4 Local Boundedness Estimates for Weak Solutions of the Heat Equation 4.4.1 Weak subsolutions (supersolutions) 4.4.2 Local boundedness estimate for weak solutions of the homogeneous heat equation 4.4.3 Local boundedness estimate for weak solutions of the nonhomogeneous heat equation 5.

Harnack's Inequalities

xi

105 105 107 Ill Ill 112 116 116 118 120 122 123 123 123 126 131

5.1 Harnack's Inequalities for Solutions of Laplace's Equation . 131 5.1.1 Mean value formula 131 5.1.2 Classical Harnack's inequality 133 5.1.3 Estimate of sup u 133 BBR

5.1.4

Estimate of inf u

135

BBR

5.1.5 Harnack's inequality 5.1.6 Holder's estimate

141 143

xii

6.

7.

8.

Elliptic and Parabolic

Equations

5.2 Harnack's Inequalities for Solutions of the Homogeneous Heat Equation 5.2.1 Weak Harnack's inequality 5.2.2 Holder's estimate 5.2.3 Harnack's inequality

145 146 155 156

Schauder's Estimates for Linear Elliptic Equations

159

6.1 Campanato Spaces 6.2 Schauder's Estimates for Poisson's Equation 6.2.1 Estimates to be established 6.2.2 Caccioppoli's inequalities 6.2.3 Interior estimate for Laplace's equation 6.2.4 Near boundary estimate for Laplace's equation . . . 6.2.5 Iteration lemma 6.2.6 Interior estimate for Poisson's equation 6.2.7 Near boundary estimate for Poisson's equation . . . 6.3 Schauder's Estimates for General Linear Elliptic Equations 6.3.1 Simplification of the problem 6.3.2 Interior estimate 6.3.3 Near boundary estimate 6.3.4 Global estimate

159 165 165 168 173 175 177 178 181 187 188 188 191 193

Schauder's Estimates for Linear Parabolic Equations

197

7.1 t-Anisotropic Campanato Spaces 7.2 Schauder's Estimates for the Heat Equation 7.2.1 Estimates to be established 7.2.2 Interior estimate 7.2.3 Near bottom estimate 7.2.4 Near lateral estimate 7.2.5 Near lateral-bottom estimate 7.2.6 Schauder's estimates for general linear parabolic equations

197 199 199 200 208 214 227

Existence of Classical Solutions for Linear Equations

233

8.1 Maximum Principle and Comparison Principle 8.1.1 The case of elliptic equations 8.1.2 The case of parabolic equations

233 233 236

231

Contents

Existence and Uniqueness of Classical Solutions for Linear Elliptic Equations 8.2.1 Existence and uniqueness of the classical solution for Poisson's equation 8.2.2 The method of continuity 8.2.3 Existence and uniqueness of classical solutions for general linear elliptic equations 8.3 Existence and Uniqueness of Classical Solutions for Linear Parabolic Equations 8.3.1 Existence and uniqueness of the classical solution for the heat equation 8.3.2 Existence and uniqueness of classical solutions for general linear parabolic equations

xiii

8.2

9.

V Estimates for Linear Equations and Existence of Strong Solutions 9.1 LP Estimates for Linear Elliptic Equations and Existence and Uniqueness of Strong Solutions 9.1.1 LP estimates for Poisson's equation in cubes 9.1.2 LP estimates for general linear elliptic equations . . . 9.1.3 Existence and uniqueness of strong solutions for linear elliptic equations 9.2 LP Estimates for Linear Parabolic Equations and Existence and Uniqueness of Strong Solutions 9.2.1 LP estimates for the heat equation in cubes 9.2.2 LP estimates for general linear parabolic equations . 9.2.3 Existence and uniqueness of strong solutions for linear parabolic equations

10. Fixed Point Method 10.1 Framework of Solving Quasilinear Equations via Fixed Point Method 10.1.1 Leray-Schauder's fixed point theorem 10.1.2 Solvability of quasilinear elliptic equations 10.1.3 Solvability of quasilinear parabolic equations 10.1.4 The procedures of the a priori estimates 10.2 Maximum Estimate 10.3 Interior Holder's Estimate

240 240 246 248 249 250 251

255 255 255 260 264 266 266 271 272 277 277 277 277 280 282 282 284

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10.4 Boundary Holder's Estimate and Boundary Gradient Estimate for Solutions of Poisson's Equation 10.5 Boundary Holder's Estimate and Boundary Gradient Estimate 10.6 Global Gradient Estimate 10.7 Holder's Estimate for a Linear Equation 10.7.1 An iteration lemma 10.7.2 Morrey's theorem 10.7.3 Holder's estimate 10.8 Holder's Estimate for Gradients 10.8.1 Interior Holder's estimate for gradients of solutions . 10.8.2 Boundary Holder's estimate for gradients of solutions 10.8.3 Global Holder's estimate for gradients of solutions . 10.9 Solvability of More General Quasilinear Equations 10.9.1 Solvability of more general quasilinear elliptic equations 10.9.2 Solvability of more general quasilinear parabolic equations 11. Topological Degree Method 11.1 Topological Degree 11.1.1 Brouwer degree 11.1.2 Leray-Schauder degree 11.2 Existence of a Heat Equation with Strong Nonlinear Source 12. Monotone Method

287 289 296 301 301 302 303 307 307 308 310 310 310 311 313 313 313 315 317 323

12.1 Monotone Method for Parabolic Problems 323 12.1.1 Definition of supersolutions and subsolutions 324 12.1.2 Iteration and monotone property 324 12.1.3 Existence results 327 12.1.4 Application to more general parabolic equations . . . 330 12.1.5 Nonuniqueness of solutions 332 12.2 Monotone Method for Coupled Parabolic Systems 336 12.2.1 Quasimonotone reaction functions 337 12.2.2 Definition of supersolutions and subsolutions 337 12.2.3 Monotone sequences 339 12.2.4 Existence results 350 12.2.5 Extension 353

Contents

13. Degenerate Equations 13.1 Linear Equations 13.1.1 Formulation of the first boundary value problem . . 13.1.2 Solvability of the problem in a space similar to Hx . 13.1.3 Solvability of the problem in IP (ft) 13.1.4 Method of elliptic regularization 13.1.5 Uniqueness of weak solutions in L p (ft) and regularity 13.2 A Class of Special Quasilinear Degenerate Parabolic Equations - Filtration Equations 13.2.1 Definition of weak solutions 13.2.2 Uniqueness of weak solutions for one dimensional equations 13.2.3 Existence of weak solutions for one dimensional equations 13.2.4 Uniqueness of weak solutions for higher dimensional equations 13.2.5 Existence of weak solutions for higher dimensional equations 13.3 General Quasilinear Degenerate Parabolic Equations . . . . 13.3.1 Uniqueness of weak solutions for weakly degenerate equations 13.3.2 Existence of weak solutions for weakly degenerate equations 13.3.3 A remark on quasilinear parabolic equations with strong degeneracy

xv

355 355 356 361 362 365 366 368 369 371 373 378 381 384 385 393 399

Bibliography

403

Index

405

Chapter 1

Preliminary Knowledge

In this chapter, we provide some preliminary knowledge needed in this book. The central part is a brief introduction to the theory of Sobolev spaces and Holder spaces. Most results are stated without proof, but references containing detailed proofs are indicated. An exception is that, for the convenience of the reader, a thorough discussion about the trace on the boundary of functions in a class of special Sobolev spaces is presented. The reader is assumed to have some acquaintance with elementary knowledge of functional analysis. Some specific facts in this field will be quoted wherever we need in the following chapters. 1.1

Some Frequently Applied Inequalities and Basic Techniques

This section presents some frequently applied inequalities and basic techniques such as mollifying, cutting off, partition of unity and local flatting of the boundary. 1.1.1

Some frequently

Young's inequality

applied

inequalities

Let a > 0, b > 0, p > 1, q > 1 and - + - = 1. P

Then L

a?

ab
1, q > 1 and - + - = 1. If f € LP(Q,),

g e Li(£l), then f • g e Ll(£l) and / \f(x)g(x)\dx Jn

< ||/(a:)||LP(n)||5(a;)|| L , ( n).

In particular, when p = q = 2, it becomes / \f(x)g(x)\dx Jo.