目录
- Contents
Series Prefaceiii
Prefacev
1 Introduction 1
1.1 Basic Mathematical Questions 2
1.1.1Existence 3
1.1.2 Multiplicity 4
1.1.3 Stability 6
1.1.4 Linear Systems of ODEs and Asymptotic Stability 7
1.1.5 Well-Posed Problems 8
1.1.6 Representations 9
1.1.7 Estimation 10
1.1.8 Smoothness 12
1.2 Elementary Partial Differential Equations 14
1.2.1 Laplace's Equation 15
1.2.2 The Heat Equation 24
1.2.3The Wave Equation 30
2 Characteristics 36
2.1Classification and Characteristics 36
2.1.1The Symbol of a Differential Expression 37
2.1.2 Scalar Equations of Second Ordcr 38
2.1.3 Higher-Order Equations and Systems 41
2.1.4 Nonlinear Equations 44
2.2 The Cauchy-Kovalevskaya Theorem 46
2.2.1 Real Analytic Functions 46
2.2.2 Majorization 50
2.2.3 Statement and Proof of the Theorem 51
2.2.4 Reduction of General Systems 53
2.2.5 A PDE without Solutions 57
2.3 Holmgren's Uniqueness Theorem 61
2.3.1 An Outline of the Main Idea 61
2.3.2 Statement and Proof of the Theorem 62
2.3.3 The Weierstra？ Approximation Theorem 64
3 Conservation Laws and Shocks 67
3.1 Systems in One Space Dimension 68
3.2 Basic Definitions and Hypotheses 70
3.3 Blowup of Smooth Solutions 73
3.3.1 Single Conservation Laws 73
3.3.2 The p System 76
3.4 Weak Solutions 77
3.4.1 The Rankine-Hugoniot Condition 79
3.4.2 Multiplicity 81
3.4.3 The Lax Shock Condition 83
3.5 Riemann Problems 84
3.5.1 Single Equations 85
3.5.2 Systems 86
3.6 Other Selection Criteria 94
3.6.1 The Entropy Condition 94
3.6.2 Viscosity Solutions 97
3.6.3 Uniqueness 99
4 Maximum Principles 101
4.1 Maximum Principles of Elliptic Problems 102
4.1.1 The Weak Maximum Principle 102
4.1.2 The Strong Maximum Principle 103
4.1.3 A Priori Bounds.105
4.2 An Existence Proof for the Dirichlet Problem 107
4.2.1 The Dirichlet Problem on a Ball 108
4.2.2 Subharmonic Functions.109
4.2.3 The Arzela-Ascoli Theorem 110
4.2 .4 Proof of Theorem 4.13 112
4.3 Radial Symmetry 114
4.3.1 Two Auxiliary Lemmas 114
4.3.2 Proof of the Theorem 115
4.4 Maximum Principles for Parabolic Equations 117
4.4 .1 The Weak Maximum Principle 117
4.4.2 The Strong Maximum Principle 118
5 Distributions 122
5.1 Test Functions and Distributions 122
5.1.1 Motivation 122
5.1.2 Test Functions 124
5.1.3 Distributions 126
5.1.4 Localization and Regularization 129
5.1.5 Convergencc of Distributions 130
5.1.6 Tempered Distributions 132
5.2 Derivatives and Integrals 135
5.2.1 Basic Definitions 135
5.2.2 Examples 136
5.2.3 Primitives and Ordinary Differential Equations 140
5.3 Convolutions and Fundamental Solutions 143
5.3.1 The Direct Product of Distributions 143
5.3.2 Convolution of Distributions 145
5.3.3 mdamental Solutions 147
5.4 The Fourier Lansform 151
5.4 .1 Fourier Transforms of Test Functions 151
5 .4 .2 Fourier Transforms of Tempered Distributions 153
5 .4 .3 The Fundamental Solution for the Wave Equation 156
5.4.4 Fourier Transform of Convolutions 158
5 .4 .5 Laplace Transforms 159
5.5 Green's Functions 163
5.5.1 Boundary-Value Problems and their Adjoints 163
5.5.2 Green's Functions for Boundary-Value Problems 167
5.5.3 Boundary Integral Methods 170
6 Fu nction Spaces 174
6.1 Banach Spaces and Hilbert Spaces 174
6.1.1 Banach Spaces.174
6.1.2 Examples of Banach Spaces 177
6.1.3 Hilbert Spaces 180
6.2 Bases in Hilbert Spaces 184
6.2.1 The Existence of a ？asis 184
6.2.2 Fourier Series 188
6.2.3 Orthogonal Polynomials 190
6.3 Duality and Weak Convergence 194
6.3.1 Bounded Linear Mappings 194
6.3.2 Examples of Dual Spaces 195
6.3.3 The Hahn-Banach Theorem 197
6.3 .4 The Uniform Boundedness Theorem 198
6.3.5 Weak Convergence 199
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 Thc Trace Theorem 214
7.3 Negative Sobolev Spaces and Duality 218
7.4 Technical Results 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 Bou日ded Inverse Theorem for Hilbert Spaces 257
8.5 Compact Operators 259
8.5.1 The Spectrum of a Compact Operator 265
8.6 Sturm-Liouvillc Boundary-Value Problems 271
8.7 The Fredholm Index 279
9 Linear Elliptic Equations 283
9.1 Definitions 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 G？rding's Inequality 292
9.2.4 Existence of Weak Solutions 298
9.3 Eigenfunction Expansions 300
9.3.1 Fredholm Theory 300
9.3.2 Eigenfunction Expansions 302
9.4 General Linear Elliptic Problems 303
9.4.1 The Neumann Problem 304
9.4.2 The Complcmenting Condition for Elliptic Systems 306
9.4.3 The Adjoint Boundary-Value Problem 311
9.4 .4 Agmon's Condition and Coercive Problerns 315
9.5 Interior Regularity 318
9.5.1 Difference Quotients 321
9.5.2 Second-Order Scalar Equations 323
9.6 Boundary Regularity 324
10 Nonlinear Elliptic Equations 335
10.1 Perturbation Results 335
10.1.1 The Banach Contraction Principle and the Implicit Fu nction 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 Nernytskii Operators 370
10.3.5 Pseudo-monotone Operators 371
10.3.6 Application to PDEs 374
11 Energy Methods for Evolution Problems 380
11.1 Parabolic Equations 380
11.1.1 Banach Space Valued Functions and Distributions 380
11.1.2 Abstract Parabolic Initial-Value Problems 382
11.1.3 Applications 385
11.1.4 Regularity of Solutions 386
11.2 Hyperbolic Evolution Problems 388
1 1.2.1 Abstract Second-Order Evolution Problems 388
11.2.2 Existence of a Solution 389
11.2.3 Uniqueness of the Solution 391
11.2.4 Continuity of the Solution 392
12 Semigroup Methods 395
12.1 Semigroups and Infinitesimal Generators 397
12.1.1 Strongly Continuous Semigroups 397
12.1.2 The Infinitesimal Generator 399
12.1.3 Abstract ODEs 401
12.2 The Hille-Yosida Theorem 403
12.2.1 The Hille-Yosida Theorem 403
12.2.2 The Lumer-Phillips Theorem 406
12.3 Applications to PDEs 408
12.3.1 Symmetric Hyperbolic Systems 408
12.3.2 The Wave Equation 410
12.3.3 The Schr？dinger Equation 411
12.4 Analytic Semigroups 413
12.4.1 Analytic Semigroups and Their Generators 413
12 .4 .2 Lactional Powers 416
12.4.3 Perturbations of Analytic Semigroups 419
12.4.4 Regularity of Mild Solutions 422
A References 426
A.1 Elementary Texts 426
A.2 Basic Graduate Texts 427
A.3 Specialized or Advanced Texts 427
A.4 Multivolume or Encyclopedic Works 429
A.5 Other References 429
Index 431
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