偏微分方程引论 影印版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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1 Introduction1

1.1 Basic Mathematical Questions2

1.1.1 Existence2

1.1.2 Multiplicity4

1.1.3 Stability6

1.1.4 Linear Systems of ODEs and Asymptotic Stability7

1.1.5 Well-Posed Problems8

1.1.6 Representations9

1.1.7 Estimation10

1.1.8 Smoothness12

1.2 Elementary Partial Differential Equations14

1.2.1 Laplace's Equation15

1.2.2 The Heat Equation24

1.2.3 The Wave Equation30

2 Characteristics36

2.1 Classification and Characteristics36

2.1.1 The Symbol of a Differential Expression37

2.1.2 Scalar Equations of Second Order38

2.1.3 Higher-Order Equations and Systems41

2.1.4 Nonlinear Equations44

2.2 The Cauchy-Kovalevskaya Theorem46

2.2.1 Real Analytic Punctions46

2.2.2 Majorization50

2.2.3 Statement and Proof of the Theorem51

2.2.4 Reduction of General Systems53

2.2.5 A PDE without Solutions57

2.3 Holmgren's Uniqueness Theorem61

2.3.1 An Outline of the Main Idea61

2.3.2 Statement and Proof of the Theorem62

2.3.3 The WeierstraβApproximation Theorem64

3 Conservation Laws and Shocks67

3.1 Systems in One Space Dimension68

3.2 Basic Definitions and Hypotheses70

3.3 Blowup of Smooth Solutions73

3.3.1 Single Conservation Laws73

3.3.2 The p System76

3.4 Weak Solutions77

3.4.1 The Rankine-Hugoniot Condition79

3.4.2 Multiplicity81

3.4.3 The Lax Shock Condition83

3.5 Riemann Problems84

3.5.1 Single Equations85

3.5.2 Systems86

3.6 Other Selection Criteria94

3.6.1 The Entropy Condition94

3.6.2 Viscosity Solutions97

3.6.3 Uniqueness99

4 Maximum Principles101

4.1 Maximum Principles of Elliptic Problems102

4.1.1 The Weak Maximum Principle102

4.1.2 The Strong Maximum Principle103

4.1.3 A Priori Bounds105

4.2 An Existence Proof for the Dirichlet Problem107

4.2.1 The Dirichlet Problem on a Ball108

4.2.2 Subharmonic Functions109

4.2.3 The Arzela-Ascoli Theorem110

4.2.4 Proof of Theorem 4.13112

4.3 Radial Symmetry114

4.3.1 Two Auxiliary Lemmas114

4.3.2 Proof of the Theorem115

4.4 Maximum Principles for Parabolic Equations117

4.4.1 The Weak Maximum Principle117

4.4.2 The Strong Maximum Principle118

5 Distributions122

5.1 Test Functions and Distributions122

5.1.1 Motivation122

5.1.2 Test Functions124

5.1.3 Distributions126

5.1.4 Localization and Regularization129

5.1.5 Convergencc of Distributions130

5.1.6 Tempered Distributions132

5.2 Derivatives and Integrals135

5.2.1 Basic Definitions135

5.2.2 Examples136

5.2.3 Primitives and Ordinary Differential Equations140

5.3 Convolutions and Fundamental Solutions143

5.3.1 The Direct Product of Distributions143

5.3.2 Convolution of Distributions145

5.3.3 Fundamental Solutions147

5.4 The Fourier Transform151

5.4.1 Fourier Transforms of Test Functions151

5.4.2 Fourier Transforms of Tempered Distributions153

5.4.3 The Fundamental Solution for the Wave Equation156

5.4.4 Fourier Transform of Convolutions158

5.4.5 Laplace Transforms159

5.5 Green's Functions163

5.5.1 Boundary-Value Problems and their Adjoints163

5.5.2 Green's Functions for Boundary-Value Problems167

5.5.3 Boundary Integral Methods170

6 Function Spaces174

6.1 Banach Spaces and Hilbert Spaces174

6.1.1 Banach Spaces174

6.1.2 Examples of Banach Spaces177

6.1.3 Hilbert Spaces180

6.2 Bases in Hilbert Spaces184

6.2.1 The Existence of a Basis184

6.2.2 Fourier Series188

6.2.3 Orthogonal Polynomials190

6.3 Duality and Weak Convergence194

6.3.1 Bounded Linear Mappings194

6.3.2 Examples of Dual Spaces195

6.3.3 The Hahn-Banach Theorem197

6.3.4 The Uniform Boundedness Theorem198

6.3.5 Weak Convergence199

7 Sobolev Spaces203

7.1 Basic Definitions204

7.2 Characterizations of Sobolev Spaces207

7.2.1 Some Comments on the DomainΩ207

7.2.2 Sobolev Spaces and Fourier Transform208

7.2.3 The Sobolev Imbedding Theorem209

7.2.4 Compactness Properties210

7.2.5 The Trace Theorem214

7.3 Negative Sobolev Spaces and Duality218

7.4 Technical Results220

7.4.1 Density Theorems220

7.4.2 Coordinate Transformations and Sobolev Spaces on Manifolds221

7.4.3 Extension Theorems223

7.4.4 Problems225

8 Operator Theory228

8.1 Basic Definitions and Examples229

8.1.1 Operators229

8.1.2 Inverse Operators230

8.1.3 Bounded Operators,Extensions230

8.1.4 Examples of Operators232

8.1.5 Closed Operators237

8.2 The Open Mapping Theorem241

8.3 Spectrum and Resolvent244

8.3.1 The Spectra of Bounded Operators246

8.4 Symmetry and Self-adjointness251

8.4.1 The Adjoint Operator251

8.4.2 The Hilbert Adjoint Operator253

8.4.3 Adjoint Operators and Spectral Theory256

8.4.4 Proof of the Bounded Inverse Theorem for Hilbert Spaces257

8.5 Compact Operators259

8.5.1 The Spectrum of a Compact Operator265

8.6 Sturm-Liouville Boundary-Value Problems271

8.7 The Fredholm Index279

9 Linear Elliptic Equations283

9.1 Definitions283

9.2 Existence and Uniqueness of Solutions of the Dirichlet Problem287

9.2.1 The Dirichlet Problem—Types of Solutions287

9.2.2 The Lax-Milgram Lemma290

9.2.3 G？rding's Inequality292

9.2.4 Existence of Weak Solutions298

9.3 Eigenfunction Expansions300

9.3.1 Fredholm Theory300

9.3.2 Eigenfunction Expansions302

9.4 General Linear Elliptic Problems303

9.4.1 The Neumann Problem304

9.4.2 The Complementing Condition for Elliptic Systems306

9.4.3 The Adjoint Boundary-Value Problem311

9.4.4 Agmon's Condition and Coercive Problems315

9.5 Interior Regularity318

9.5.1 Difference Quotients321

9.5.2 Second-Order Scalar Equations323

9.6 Boundary Regularity324

10 Nonlinear Elliptic Equations335

10.1 Perturbation Results335

10.1.1 The Banach Contraction Principle and the Implicit Function Theorem336

10.1.2 Applications to Elliptic PDEs339

10.2 Nonlinear Variational Problems342

10.2.1 Convex problems342

10.2.2 Nonconvex Problems355

10.3 Nonlinear Operator Theory Methods359

10.3.1 Mappings on Finite-Dimensional Spaces359

10.3.2 Monotone Mappings on Banach Spaces363

10.3.3 Applications of Monotone Operators to Nonlinear PDEs366

10.3.4 Nemytskii Operators370

10.3.5 Pseudo-monotone Operators371

10.3.6 Application to PDEs374

11 Energy Methods for Evolution Problems380

11.1 Parabolic Equations380

11.1.1 Banach Space Valued Functions and Distributions380

11.1.2 Abstract Parabolic Initial-Value Problems382

11.1.3 Applications385

11.1.4 Regularity of Solutions386

11.2 Hyperbolic Evolution Problems388

11.2.1 Abstract Second-Order Evolution Problems388

11.2.2 Existence of a Solution389

11.2.3 Uniqueness of the Solution391

11.2.4 Continuity of the Solution392

12 Semigroup Methods395

12.1 Semigroups and Infinitesimal Generators397

12.1.1 Strongly Continuous Semigroups397

12.1.2 The Infinitesimal Generator399

12.1.3 Abstract ODEs401

12.2 The Hille-Yosida Theorem403

12.2.1 The Hille-Yosida Theorem403

12.2.2 The Lumer-Phillips Theorem406

12.3 Applications to PDEs408

12.3.1 Symmetric Hyperbolic Systems408

12.3.2 The Wave Equation410

12.3.3 The Schr？dinger Equation411

12.4 Analytic Semigroups413

12.4.1 Analytic Semigroups and Their Generators413

12.4.2 Fractional Powers416

12.4.3 Perturbations of Analytic Semigroups419

12.4.4 Regularity of Mild Solutions422

A References426

A.1 Elementary Texts426

A.2 Basic Graduate Texts427

A.3 Specialized or Advanced Texts427

A.4 Multivolume or Encyclopedic Works429

A.5 Other Refcrences429

Index431