属Springer Texts in Applied Mathematics 系列，研究生PDE入门教材，对PDE作了全面介绍。

目录如下：
1 Introduction 1
1.1 BasicMathematical Questions . . . . . . . . . . . . . . . 2
1.1.1 Existence . . . . . . . . . . . . . . . . . . . . . . . 2
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
1.2.3 TheWave Equation . . . . . . . . . . . . . . . . . 30
2 Characteristics 36
2.1 Classification and Characteristics . . . . . . . . . . . . . 36
2.1.1 The Symbol of a Differential Expression . . . . . 37
2.1.2 Scalar Equations of Second Order . . . . . . . . . 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 theMain Idea . . . . . . . . . . . . 61
2.3.2 Statement and Proof of the Theorem . . . . . . . 62
2.3.3 TheWeierstra┈ 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 TheWeakMaximumPrinciple . . . . . . . . . . . 102
4.1.2 The StrongMaximumPrinciple . . . . . . . . . . 103
4.1.3 A Priori Bounds . . . . . . . . . . . . . . . . . . . 105
4.2 An Existence Proof for the Dirichlet Problem . . . . . . 107
4.2.1 The Dirichlet Problemon a Ball . . . . . . . . . . 108
4.2.2 Subharmonic Functions . . . . . . . . . . . . . . . 109
4.2.3 The Arzela-Ascoli Theorem . . . . . . . . . . . . 110
4.2.4 Proof of Theorem4.13 . . . . . . . . . . . . . . . 112
4.3 Radial Symmetry . . . . . . . . . . . . . . . . . . . . . . 114
4.3.1 Two Auxiliary Lemmas . . . . . . . . . . . . . . . 114
4.3.2 Proof of the Theorem . . . . . . . . . . . . . . . . 115
4.4 MaximumPrinciples for Parabolic Equations . . . . . . . 117
4.4.1 TheWeakMaximumPrinciple . . . . . . . . . . . 117
4.4.2 The StrongMaximumPrinciple . . . . . . . . . . 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 Convergence 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 Fundamental Solutions . . . . . . . . . . . . . . . 147
5.4 The Fourier Transform . . . . . . . . . . . . . . . . . . . 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 Transformof 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 Function 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 Basis . . . . . . . . . . . . . . 184
6.2.2 Fourier Series . . . . . . . . . . . . . . . . . . . . 188
6.2.3 Orthogonal Polynomials . . . . . . . . . . . . . . 190
6.3 Duality andWeak 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 ヘ . . . . . . . . 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 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 OpenMapping Theorem . . . . . . . . . . . . . . . 241
8.3 Spectrumand 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 FredholmIndex . . . . . . . . . . . . . . . . . . . . 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-MilgramLemma . . . . . . . . . . . . . 290
9.2.3 Gⅹarding’s Inequality . . . . . . . . . . . . . . . . 292
9.2.4 Existence ofWeak Solutions . . . . . . . . . . . . 298
9.3 Eigenfunction Expansions . . . . . . . . . . . . . . . . . 300
9.3.1 FredholmTheory . . . . . . . . . . . . . . . . . . 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 Complementing Condition for Elliptic Systems 306
9.4.3 The Adjoint Boundary-Value Problem . . . . . . 311
9.4.4 Agmon’s Condition and Coercive Problems . . . . 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
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