Elliptic And Parabolic Equations

by Zhuoqun Wu, Jingxue Yin, Chunpeng Wang

Overview

This book provides an introduction to elliptic and parabolic equations. While there are numerous monographs focusing separately on each kind of equations, there are very few books treating these two kinds of equations in combination. This book presents the related basic theories and methods to enable readers to appreciate the commonalities between these two kinds of equations as well as contrast the similarities and differences between them.

Product Details

ISBN-13:	9789812700254
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/18/2006
Pages:	424
Product dimensions:	6.20(w) x 9.00(h) x 1.00(d)

Table of Contents

Preface     v
Preliminary Knowledge     1
Some Frequently Applied Inequalities and Basic Techniques     1
Some frequently applied inequalities     1
Spaces C[superscript k]([Omega]) and C[Characters not reproducible]([Omega])     2
Smoothing operators     3
Cut-off functions     5
Partition of unity     6
Local flatting of the boundary     6
Holder Spaces     7
Spaces C[superscript k, alpha]([Characters not reproducible]) and C[superscript k, alpha]([Omega])     7
Interpolation inequalities     8
Spaces C[superscript 2k+alpha,k+alpha/2]([Characters not reproducible][subscript T])     13
Isotropic Sobolev Spaces     14
Weak derivatives     14
Sobolev spaces W[superscript k,p]([Omega]) and W[Characters not reproducible]([Omega])     15
Operation rules of weak derivatives     17
Interpolation inequality     17
Embedding theorem     19
Poincare's inequality     21
t-Anisotropic Sobolev Spaces     24
Spaces W[Characters not reproducible] (Q[subscript T]), W[Characters not reproducible](Q[subscript T]), W[Characters not reproducible]), V[subscript 2](Q[subscript T]) and V(Q[subscript T])     24
Embedding theorem     26
Poincare's inequality     28
Trace of Functions in H[superscript 1]([Omega])     29
Some propositions on functions in H[superscript 1](Q[superscript +])     29
Trace of functions in H[superscript 1]([Omega])     33
Trace of functions in H[superscript 1](Q[subscript T]) = W[Characters not reproducible](Q[superscript T])     35
L[superscript 2] Theory of Linear Elliptic Equations     39
Weak Solutions of Poisson's Equation     39
Definition of weak solutions     40
Riesz's representation theorem and its application     41
Transformation of the problem     43
Existence of minimizers of the corresponding functional     44
Regularity of Weak Solutions of Poisson's Equation     47
Difference operators     47
Interior regularity     50
Regularity near the boundary     53
Global regularity     56
Study of regularity by means of smoothing operators     58
L[superscript 2] Theory of General Elliptic Equations     60
Weak solutions     60
Riesz's representation theorem and its application     61
Variational method     62
Lax-Milgram's theorem and its application      64
Fredholm's alternative theorem and its application     67
L[superscript 2] Theory of Linear Parabolic Equations     71
Energy Method     71
Definition of weak solutions     72
A modified Lax-Milgram's theorem     73
Existence and uniqueness of the weak solution     75
Rothe's Method     79
Galerkin's Method     85
Regularity of Weak Solutions     89
L[superscript 2] Theory of General Parabolic Equations     94
Energy method     94
Rothe's method     96
Galerkin's method     97
De Giorgi Iteration and Moser Iteration     105
Global Boundedness Estimates of Weak Solutions of Poisson's Equation     105
Weak maximum principle for solutions of Laplace's equation     105
Weak maximum principle for solutions of Poisson's equation     107
Global Boundedness Estimates for Weak Solutions of the Heat Equation     111
Weak maximum principle for solutions of the homogeneous heat equation     111
Weak maximum principle for solutions of the nonhomogeneous heat equation     112
Local Boundedness Estimates for Weak Solutions of Poisson's Equation     116
Weak subsolutions (supersolutions)      116
Local boundedness estimate for weak solutions of Laplace's equation     118
Local boundedness estimate for solutions of Poisson's equation     120
Estimate near the boundary for weak solutions of Poisson's equation     122
Local Boundedness Estimates for Weak Solutions of the Heat Equation     123
Weak subsolutions (supersolutions)     123
Local boundedness estimate for weak solutions of the homogeneous heat equation     123
Local boundedness estimate for weak solutions of the nonhomogeneous heat equation     126
Harnack's Inequalities     131
Harnack's Inequalities for Solutions of Laplace's Equation     131
Mean value formula     131
Classical Harnack's inequality     133
Estimate of [Characters not reproducible] u     133
Estimate of [Characters not reproducible] u     135
Harnack's inequality     141
Holder's estimate     143
Harnack's Inequalities for Solutions of the Homogeneous Heat Equation     145
Weak Harnack's inequality     146
Holder's estimate     155
Harnack's inequality     156
Schauder's Estimates for Linear Elliptic Equations     159
Campanato Spaces     159
Schauder's Estimates for Poisson's Equation     165
Estimates to be established     165
Caccioppoli's inequalities     168
Interior estimate for Laplace's equation     173
Near boundary estimate for Laplace's equation     175
Iteration lemma     177
Interior estimate for Poisson's equation     178
Near boundary estimate for Poisson's equation     181
Schauder's Estimates for General Linear Elliptic Equations     187
Simplification of the problem     188
Interior estimate     188
Near boundary estimate     191
Global estimate     193
Schauder's Estimates for Linear Parabolic Equations     197
t-Anisotropic Campanato Spaces     197
Schauder's Estimates for the Heat Equation     199
Estimates to be established     199
Interior estimate     200
Near bottom estimate     208
Near lateral estimate     214
Near lateral-bottom estimate     227
Schauder's estimates for general linear parabolic equations     231
Existence of Classical Solutions for Linear Equations     233
Maximum Principle and Comparison Principle     233
The case of elliptic equations      233
The case of parabolic equations     236
Existence and Uniqueness of Classical Solutions for Linear Elliptic Equations     240
Existence and uniqueness of the classical solution for Poisson's equation     240
The method of continuity     246
Existence and uniqueness of classical solutions for general linear elliptic equations     248
Existence and Uniqueness of Classical Solutions for Linear Parabolic Equations     249
Existence and uniqueness of the classical solution for the heat equation     250
Existence and uniqueness of classical solutions for general linear parabolic equations     251
L[superscript p] Estimates for Linear Equations and Existence of Strong Solutions     255
L[superscript p] Estimates for Linear Elliptic Equations and Existence and Uniqueness of Strong Solutions     255
L[superscript p] estimates for Poisson's equation in cubes     255
L[superscript p] estimates for general linear elliptic equations     260
Existence and uniqueness of strong solutions for linear elliptic equations     264
L[superscript p] Estimates for Linear Parabolic Equations and Existence and Uniqueness of Strong Solutions     266
L[superscript p] estimates for the heat equation in cubes     266
L[superscript p] estimates for general linear parabolic equations      271
Existence and uniqueness of strong solutions for linear parabolic equations     272
Fixed Point Method     277
Framework of Solving Quasilinear Equations via Fixed Point Method     277
Leray-Schauder's fixed point theorem     277
Solvability of quasilinear elliptic equations     277
Solvability of quasilinear parabolic equations     280
The procedures of the a priori estimates     282
Maximum Estimate     282
Interior Holder's Estimate     284
Boundary Holder's Estimate and Boundary Gradient Estimate for Solutions of Poisson's Equation     287
Boundary Holder's Estimate and Boundary Gradient Estimate     289
Global Gradient Estimate     296
Holder's Estimate for a Linear Equation     301
An iteration lemma     301
Morrey's theorem     302
Holder's estimate     303
Holder's Estimate for Gradients     307
Interior Holder's estimate for gradients of solutions     307
Boundary Holder's estimate for gradients of solutions     308
Global Holder's estimate for gradients of solutions     310
Solvability of More General Quasilinear Equations     310
Solvability of more general quasilinear elliptic equations      310
Solvability of more general quasilinear parabolic equations     311
Topological Degree Method     313
Topological Degree     313
Brouwer degree     313
Leray-Schauder degree     315
Existence of a Heat Equation with Strong Nonlinear Source     317
Monotone Method     323
Monotone Method for Parabolic Problems     323
Definition of supersolutions and subsolutions     324
Iteration and monotone property     324
Existence results     327
Application to more general parabolic equations     330
Nonuniqueness of solutions     332
Monotone Method for Coupled Parabolic Systems     336
Quasimonotone reaction functions     337
Definition of supersolutions and subsolutions     337
Monotone sequences     339
Existence results     350
Extension     353
Degenerate Equations     355
Linear Equations     355
Formulation of the first boundary value problem     356
Solvability of the problem in a space similar to H[superscript 1]     361
Solvability of the problem in L[superscript p]([Omega])     362
Method of elliptic regularization     365
Uniqueness of weak solutions in L[superscript p]([Omega]) and regularity     366
A Class of Special Quasilinear Degenerate Parabolic Equations - Filtration Equations     368
Definition of weak solutions     369
Uniqueness of weak solutions for one dimensional equations     371
Existence of weak solutions for one dimensional equations     373
Uniqueness of weak solutions for higher dimensional equations     378
Existence of weak solutions for higher dimensional equations     381
General Quasilinear Degenerate Parabolic Equations     384
Uniqueness of weak solutions for weakly degenerate equations     385
Existence of weak solutions for weakly degenerate equations     393
A remark on quasilinear parabolic equations with strong degeneracy     399
Bibliography     403
Index     405