线性偏微分算子分析 第2卷PDF电子书下载

Introduction 1

Chapter Ⅹ．Existence and Approximation of Solutions of Differential Equations 3

Summary 3

10.1．The Spaces Bp,k 3

10.2．Fundamental Solutions 16

10.3．The Equation P（D）u=f when f∈？′ 29

10.4．Comparison of Differential Operators 32

10.5．Approximation of Solutions of Homogeneous Differential Equations 39

10.6．The Equation P（D）u=f when f is in a Local Space ？′F 41

10.7．The Equation P（D）u=f when f∈？′（X） 45

10.8．The Geometrical Meaning of the Convexity Conditions 50

Notes 58

Chapter Ⅺ．Interior Regularity of Solutions of Differential Equations 60

Summary 60

11.1．Hypoelliptic Operators 61

11.2．Partially Hypoelliptic Operators 69

11.3．Continuation of Differentiability 73

11.4．Estimates for Derivatives of High Order 85

Notes 92

Chapter Ⅻ．The Cauchy and Mixed Problems 94

Summary 94

12.1．The Cauchy Problem for the Wave Equation 96

12.2．The Oscillatory Cauchy Problem for the Wave Equation 104

12.3．Necessary Conditions for Existence and Uniqueness of Solutions to the Cauchy Problem 110

12.4．Properties of Hyperbolic Polynomials 112

12.5．The Cauchy Problem for a Hyperbolic Equation 120

12.6．The Singularities of the Fundamental Solution 125

12.7．A Global Uniqueness Theorem 133

12.8．The Characteristic Cauchy Problem 143

12.9．Mixed Problems 162

Notes 180

Chapter ⅩⅢ．Differential Operators of Constant Strength 182

Summary 182

13.1．Definitions and Basic Properties 182

13.2．Existence Theorems when the Coefficients are Merely Continuous 184

13.3．Existence Theorems when the Coefficients are in C∞ 186

13.4．Hypoellipticity 191

13.5．Global Existence Theorems 194

13.6．Non-uniqueness for the Cauchy Problem 201

Notes 224

Chapter ⅩⅣ．Scattering Theory 225

Summary 225

14.1．Some Function Spaces 227

14.2．Division by Functions with Simple Zeros 232

14.3．The Resolvent of the Unperturbed Operator 237

14.4．Short Range Perturbations 243

14.5．The Boundary Values of the Resolvent and the Point Spectrum 251

14.6．The Distorted Fourier Transforms and the Continuous Spectrum 255

14.7．Absence of Embedded Eigenvalues 264

Notes 268

Chapter ⅩⅤ．Analytic Function Theory and Differential Equations 270

Summary 270

15.1．The Inhomogeneous Cauchy-Riemann Equations 271

15.2．The Fourier-Laplace Transform of Bc2,k（X）when X is Convex 279

15.3．Fourier-Laplace Representation of Solutions of Differential Equations 287

15.4．The Fourier-Laplace Transform of C？（X）when X is Convex 296

Notes 300

Chapter ⅩⅥ．Convolution Equations 302

Summary 302

16.1．Subharmonic Functions 303

16.2．Plurisubharmonic Functions 314

16.3．The Support and Singular Support of a Convolution 319

16.4．The Approximation Theorem 335

16.5．The Inhomogeneous Convolution Equation 341

16.6．Hypoelliptic Convolution Equations 353

16.7．Hyperbolic Convolution Equations 356

Notes 360

Appendix A．Some Algebraic Lemmas 362

A.1．The Zeros of Analytic Functions 362

A.2．Asymptotic Properties of Algebraic Functions of Several Variables 364

Notes 371

Bibliography 373

Index 391

Index of Notation 392