Hilbert Space Methods in Partial Differential Equations

Hilbert Space Methods in Partial Differential Equations

by Ralph E. Showalter

Paperback

$12.95
View All Available Formats & Editions
Eligible for FREE SHIPPING
  • Want it by Thursday, September 27  Order now and choose Expedited Shipping during checkout.

Overview

Hilbert Space Methods in Partial Differential Equations by Ralph E. Showalter

This text surveys the principal methods of solving partial differential equations. Suitable for graduate students of mathematics, engineering, and physical sciences, it requires knowledge of advanced calculus.
The initial chapter contains an elementary presentation of Hilbert space theory that provides sufficient background for understanding the rest of the book. Succeeding chapters introduce distributions and Sobolev spaces and examine boundary value problems, first- and second-order evolution equations, implicit evolution equations, and topics related to optimization and approximation. The text, which features 40 examples and 200 exercises, concludes with suggested readings and a bibliography.

Product Details

ISBN-13: 9780486474434
Publisher: Dover Publications
Publication date: 03/18/2010
Series: Dover Books on Mathematics
Pages: 224
Product dimensions: 5.90(w) x 8.90(h) x 0.50(d)

Table of Contents

I Elements of Hilbert Space 1

1 Linear Algebra 1

2 Convergence and Continuity 6

3 Completeness 10

4 Hilbert Space 14

5 Dual Operators; Identifications 19

6 Uniform Boundedness; Weak Compactness 22

7 Expansion in Eigenfunctions 24

II Distributions and Sobolev Spaces 31

1 Distributions 31

2 Sobolev Spaces 40

3 Trace 45

4 Sobolev's Lemma and Imbedding 48

5 Density and Compactness 51

III Boundary Value Problems 59

1 Introduction 59

2 Forms, Operators and Green's Formula 61

3 Abstract Boundary Value Problems 65

4 Examples 67

5 Coercivity; Elliptic Forms 74

6 Regularity 77

7 Closed operators, adjoints and eigenfunction expansions 83

IV First Order Evolution Equations 95

1 Introduction 95

2 The Cauchy Problem 98

3 Generation of Semigroups 100

4 Accretive Operators; two examples 105

5 Generation of Groups; a wave equation 109

6 Analytic Semigroups 113

7 Parabolic Equations 119

V Implicit Evolution Equations 127

1 Introduction 127

2 Regular Equations 128

3 Pseudoparabolic Equations 132

4 Degenerate Equations 136

5 Examples 138

VI Second Order Evolution Equations 145

1 Introduction 145

2 Regular Equations 146

3 Sobolev Equations 154

4 Degenerate Equations 156

5 Examples 160

VII Optimization and Approximation Topics 169

1 Dirichlet's Principle 169

2 Minimization of Convex Functions 170

3 Variational Inequalities 176

4 Optimal Control of Boundary Value Problems 180

5 Approximation of Elliptic Problems 187

6 Approximation of Evolution Equations 195

VIII Suggested Readings 207

Bibliography 209

Customer Reviews

Most Helpful Customer Reviews

See All Customer Reviews