Hilbert Space Methods in Partial Differential Equations

by Showalter, Ralph E.

• ISBN13: 9780486474434
• ISBN10: 0486474437
• Format: Paperback
• Copyright: 2010-03-18
• Publisher: Dover Publications

Summary

This text surveys the principal methods of solving partial differential equations. Suitable for graduate students of mathematics, engineering, and physical sciences, it opens with an elementary presentation of Hilbert space theory sufficient for understanding the rest of the book. Additional topics include boundary value problems, evolution equations, optimization, and approximation.1979 edition.

Table of Contents

Elements of Hilbert Space	p. 1
Linear Algebra	p. 1
Convergence and Continuity	p. 6
Completeness	p. 10
Hilbert Space	p. 14
Dual Operators; Identifications	p. 19
Uniform Boundedness; Weak Compactness	p. 22
Expansion in Eigenfunctions	p. 24
Distributions and Sobolev Spaces	p. 31
Distributions	p. 31
Sobolev Spaces	p. 40
Trace	p. 45
Sobolev's Lemma and Imbedding	p. 48
Density and Compactness	p. 51
Boundary Value Problems	p. 59
Introduction	p. 59
Forms, Operators and Green's Formula	p. 61
Abstract Boundary Value Problems	p. 65
Examples	p. 67
Coercivity; Elliptic Forms	p. 74
Regularity	p. 77
Closed operators, adjoints and eigenfunction expansions	p. 83
First Order Evolution Equations	p. 95
Introduction	p. 95
The Cauchy Problem	p. 98
Generation of Semigroups	p. 100
Accretive Operators; two examples	p. 105
Generation of Groups; a wave equation	p. 109
Analytic Semigroups	p. 113
Parabolic Equations	p. 119
Implicit Evolution Equations	p. 127
Introduction	p. 127
Regular Equations	p. 128
Pseudoparabolic Equations	p. 132
Degenerate Equations	p. 136
Examples	p. 138
Second Order Evolution Equations	p. 145
Introduction	p. 145
Regular Equations	p. 146
Sobolev Equations	p. 154
Degenerate Equations	p. 156
Examples	p. 160
Optimization and Approximation Topics	p. 169
Dirichlet's Principle	p. 169
Minimization of Convex Functions	p. 170
Variational Inequalities	p. 176
Optimal Control of Boundary Value Problems	p. 180
Approximation of Elliptic Problems	p. 187
Approximation of Evolution Equations	p. 195
Suggested Readings	p. 207
Bibliography	p. 209
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