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Elliptic and Parabolic Equations by Zhuoqun Wu, Jingxue Yin, Chunpeng Wang
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(Paperback)
- Pub. Date: October 2006
- 408pp
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Product Details
- Pub. Date: October 2006
- Publisher: World Scientific Publishing Company, Incorporated
- Format: Paperback, 408pp
- ISBN-13: 9789812700261
- ISBN: 9812700269
Synopsis
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.
From the Publisher
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
- Pub. Date: October 2006
- Publisher: World Scientific Publishing Company, Incorporated
- Format: Paperback, 408pp
- ISBN-13: 9789812700261
- ISBN: 9812700269
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
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
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