Green's Functions and Boundary Value Problems, Third Edition

880 Pages·2011·10.11 MB·English

by  Ivar Stakgold

GREEN'S FUNCTIONS
AND BOUNDARY VALUE
PROBLEMS
PURE AND APPLIED MATHEMATICS
A Wiley Series of Texts, Monographs, and Tracts
Founded by RICHARD COURANT
Editors Emeriti: MYRON B. ALLEN III, DAVID A. COX, PETER HILTON,
HARRY HOCHSTADT, PETER LAX, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
GREEN'S FUNCTIONS
AND BOUNDARY VALUE
PROBLEMS
Third Edition
Ivar Stakgold
Department of Mathematical Sciences
University of Delaware
Newark, Delaware
and
Department of Mathematics
University of California, San Diego
La Jolla, California
Michael Hoist
Departments of Mathematics and Physics
University of California, San Diego
La Jolla, California
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Library of Congress Cataloging-in-Publication Data:
Stakgold, Ivar.
Green's functions and boundary value problems / Ivar Stakgold, Michael Hoist. — 3rd ed.
p. cm. — (Pure and applied mathematics ; 99)
Includes bibliographical references and index.
ISBN 978-0-470-60970-5 (hardback)
1. Boundary value problems. 2. Green's functions. 3. Mathematical physics. I. Hoist,
Michael. II. Title.
QA379.S72 2010
515'.35—dc22 2010023290
Printed in the United States of America.
10 9 8 7 6 5 4 3 21
To Lainie andAlissa
IS.
For Mai,
Mason, and Makenna
M.H.

CONTENTS
Preface to Third Edition xi
Preface to Second Edition xv
Preface to First Edition xvii
0 Preliminaries 1
0.1 Heat Conduction 3
0.2 Diffusion 9
0.3 Reaction-Diffusion Problems 12
0.4 The Impulse-Momentum Law: The Motion of Rods and Strings 18
0.5 Alternative Formulations of Physical Problems 30
0.6 Notes on Convergence 36
0.7 The Lebesgue Integral 41
1 Green's Functions (Intuitive Ideas) 51
1.1 Introduction and General Comments 51
1.2 The Finite Rod 60
1.3 The Maximum Principle 72
1.4 Examples of Green's Functions 76
vii
VIII CONTENTS
The Theory of Distributions 91
2.1 Basic Ideas, Definitions, and Examples 91
2.2 Convergence of Sequences and Series of Distributions 110
2.3 Fourier Series 127
2.4 Fourier Transforms and Integrals 145
2.5 Differential Equations in Distributions 164
2.6 Weak Derivatives and Sobolev Spaces 181
One-Dimensional Boundary Value Problems 185
3.1 Review 185
3.2 Boundary Value Problems for Second-Order Equations 191
3.3 Boundary Value Problems for Equations of Order p 202
3.4 Alternative Theorems 206
3.5 Modified Green's Functions 216
Hilbert and Banach Spaces 223
4.1 Functions and Transformations 223
4.2 Linear Spaces 227
4.3 Metric Spaces, Normed Linear Spaces, and Banach Spaces 234
4.4 Contractions and the Banach Fixed-Point Theorem 245
4.5 Hilbert Spaces and the Projection Theorem 261
4.6 Separable Hilbert Spaces and Orthonormal Bases 275
4.7 Linear Functionals and the Riesz Representation Theorem 288
4.8 The Hahn-Banach Theorem and Reflexive Banach Spaces 292
Operator Theory 299
5.1 Basic Ideas and Examples 299
5.2 Closed Operators 307
5.3 Invertibility: The State of an Operator 311
5.4 Adjoint Operators 316
5.5 Solvability Conditions 321
5.6 The Spectrum of an Operator 326
5.7 Compact Operators 336
5.8 Extremal Properties of Operators 339
5.9 The Banach-Schauder and Banach-Steinhaus Theorems 347
CONTENTS IX
6 Integral Equations 351
6.1 Introduction 351
6.2 Fredholm Integral Equations 359
6.3 The Spectrum of a Self-Adjoint Compact Operator 370
6.4 The Inhomogeneous Equation 379
6.5 Vanational Principles and Related Approximation Methods 395
7 Spectral Theory of Second-Order Differential Operators 409
7.1 Introduction; The Regular Problem 409
7.2 Weyl's Classification of Singular Problems 432
7.3 Spectral Problems with a Continuous Spectrum 444
8 Partial Differential Equations 459
8.1 Classification of Partial Differential Equations 459
8.2 Well-Posed Problems for Hyperbolic and Parabolic Equations 472
8.3 Elliptic Equations 489
8.4 Vanational Principles for Inhomogeneous Problems 514
8.5 The Lax-Milgram Theorem 551
9 Nonlinear Problems 557
9.1 Introduction and Basic Fixed-Point Techniques 557
9.2 Branching Theory 576
9.3 Perturbation Theory for Linear Problems 584
9.4 Techniques for Nonlinear Problems 594
9.5 The Stability of the Steady State 623
10 Approximation Theory and Methods 637
10.1 Nonlinear Analysis Tools for Banach Spaces 640
10.2 Best and Near-Best Approximation in Banach Spaces 669
10.3 Overview of Sobolev and Besov Spaces 691
10.4 Applications to Nonlinear Elliptic Equations 710
10.5 Finite Element and Related Discretization Methods 736
10.6 Iterative Methods for Discretized Linear Equations 769
10.7 Methods for Nonlinear Equations 810