Modern Methods in Partial Differential Equations

by Martin Schechter

Overview

Upon its initial 1977 publication, this volume made recent accomplishments in its field available to advanced undergraduates and beginning graduate students of mathematics. Requiring only some familiarity with advanced calculus and rudimentary complex function theory, it covered discoveries of the previous three decades, a particularly fruitful era. Now it remains a permanent, much-cited contribution to the ever-expanding literature on partial differential equations.
Author Martin Schechter chose subjects that will motivate students and introduce them to techniques with wide applicability to problems in partial differential equations as well as other branches of analysis. Uniform in theme and outlook, the text features problems that consider existence, uniqueness, estimates, and regularity of solutions. Topics include existence of solutions, regularity of constant and variable coefficients, the Cauchy problem, properties of solutions, boundary value problems in a half-space, the Dirichlet problem, general domains, and general boundary value problems.

Product Details

ISBN-13:	9780486783079
Publisher:	Dover Publications
Publication date:	11/12/2013
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	256
File size:	27 MB
Note:	This product may take a few minutes to download.

About the Author

Martin Schechter is Professor of Mathematics at the University of California, Irvine. He is the author of many research papers in mathematics and Dover's Operator Methods in Quantum Mechanics.

Table of Contents

List of Symbols viii

Preface ix

1 Existence of Solutions 1

1-1 Introduction 1

1-2 Equations Without Solutions 3

1-3 Integration by Parts 7

1-4 A Necessary Condition 10

1-5 Some Notions From Hilbert Space 11

1-6 Weak Solutions 22

1-7 Operators With Constant Coefficients 24

Problems 27

2 Regularity (Constant Coefficients) 28

2-1 A Necessary Condition 28

2-2 The Friedrichs Mollifier 31

2-3 A Family of Norms 33

2-4 Elliptic Operators 37

2-5 Fourier Transforms 40

2-6 Hypoelliptic Operators 45

2-7 Comparison of Operators 46

2-8 Proof of Regularity 49

2-9 The Closed Graph Theorem 51

Problems 53

3 Regularity (Variable Coefficients) 55

3-1 Formally Hypoelliptic Operators 55

3-2 Proof of Regularity 57

3-3 Vector Spaces 62

3-4 Proof of the Lemmas 65

3-5 Existence 69

3-6 Examples 71

Problems 71

4 The Cauchy Problem 72

4-1 Statement of the Problem 72

4-2 Weak Solutions 73

4-3 Hyperbolic Equations 76

4-4 Properties of Hyperbolic Operators 80

4-5 Ordinary Differential Equations 87

4-6 Existence of Solutions 90

4-7 Uniqueness 94

Problems 98

5 Properties of Solutions 99

5-1 Existence of Strong Solutions 99

5-2 Properties of Strong Solutions 102

5-3 Estimates in One Dimension 104

5-4 Estimates in n + 1 Dimensions 111

5-5 Existence Theorems 114

5-6 Properly Hyperbolic Operators 119

5-7 Examples 120

Problems 121

6 Boundary Value Problems in a Half-Space (Elliptic) 123

6-1 Introduction 123

6-2 The Problem in a Half-Line 124

6-3 Uniqueness 128

6-4 General Boundary Conditions 130

6-5 Estimates for a Simple Case 132

6-6 Estimates for the General Case 136

6-7 Estimates in a Half-Space 139

6-8 Existence in a Half-Space 147

6-9 Some Observations 150

Problems 151

7 Boundary Value Problems in a Half-Space (Non-Elliptic) 152

7-1 Introduction 152

7-2 The Estimate in a Half-Line 152

7-3 Proof of Theorem 7-1 156

7-4 Hermitian Operators and Matrices 159

7-5 Proof of the Lemmas 163

7-6 Existence and Estimates in a Half-Space 165

7-7 Examples 168

7-8 Nonvanishing Boundary Conditions 171

Problems 174

8 The Dirichlet Problem 175

8-1 Introduction 176

8-2 A Weak Solution 176

8-3 Normal Boundary Operators 178

8-4 The Estimate 181

8-5 Compact Operators 185

8-6 Compact Embedding 186

8-7 Solving the Problem 193

8-8 Some Theorems in Half-Space 194

8-9 Regularity at the Boundary 198

Problems 201

9 General Domains 203

9-1 The Basic Theorem 203

9-2 An Inequality and a Regularity Theorem 205

9-3 Localization 209

9-4 Some Lemmas 211

9-5 The Inequality 212

9-6 Strongly Elliptic Operators 214

9-7 Garding's Inequality 216

9-8 Strong and Weak Solutions 218

9-9 The Exceptional Set 219

Problems 221

10 General Boundary Value Problems 223

10-1 Formulation of the Problem 223

10-2 The Problem in σ_R 225

10-3 The Solution 228

10-4 The Adjoint System 230

10-5 The Regularity Theorem 233

10-6 The Inequality 235

10-7 The Global Adjoint Operators 235

10-8 The Boundary Norm 237

10-9 The Compactness Argument 239

Problems 241

Bibliography 243

Subject Index 245