stak 3rd Ed contents and preface

i PURE AND APPLIED MATHEMATICS: A WILEY SERIES OF TEXTS, MONOGRAPHS, AND TRACTS Third Edition Green’s Functions and Boundary Value Problems Ivar Stakgold • Michael Holst GREEN’S FUNCTIONS AND BOUNDARY VALUE PROBLEMS GREEN’S FUNCTIONS AND BOUNDARY VALUE PROBLEMS Third Edition Ivar Stakgold Department of Mathematical Sciences University of Delaware Newark, DE –and– Department of Mathematics University of California, San Diego La Jolla, CA Michael Holst Departments of Mathematics and Physics University of California, San Diego La Jolla, CA A JOHN WILEY & SONS, INC., PUBLICATION Copyright c/circlecopyrt2011 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. 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Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data: Green’s Functions and Boundary Value Problems / Ivar Stakgold and Michael Holst p. cm.—(Wiley series in XXX) “Wiley-Interscience.” Includes bibliographical references and index. ISBN X-XXX-XXXXX-X Series. Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 To Lainie and Alissa. -I.S. 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 2 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 3 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 4 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 5 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 Variational 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 Variational 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 Index 845 PREFACE TO THE THIRD EDITION Why a third edition? The principal reason is to include more material from anal- ysis, approximation theory, partial differential equations, and numerical analysis as needed for understanding modern computational methods that play such a vital role in the solution of boundary value problems. As I am not an expert in computational mathematics, it was essential to find a highly qualified coauthor. When I moved to San Diego in early 2008, I was offered an office at the University of California, San Diego (UCSD), which, luckily, was next to the office of Michael Holst. Here was the perfect coauthor, and it was my good fortune that he agreed to collaborate on the new edition! The most substantial change for the new third edition is a fairly extensive new chapter (Chapter 10), which covers the new material listed above. The sections of the new chapter are: 10.1 Nonlinear Analysis Tools for Banach Spaces 10.2 Best and Near-Best Approximation in Banach Spaces 10.3 Overview of Sobolev and Besov Spaces 10.4 Applications to Nonlinear Elliptic Equations 10.5 Finite Element and Related Discretization Methods 10.6 Iterative Methods for Discretized Linear Equations 10.7 Methods for Nonlinear Equations To support the inclusion of this new chapter, and to help connect the presentation of the analysis material to standard references, we have added an additional final xi xii PREFACE TO THIRD EDITION section to four of the chapters that appeared in the second edition of the book. These completely new sections for the third edition are: 2.6 Weak Derivatives and Sobolev Spaces 4.8 The Hahn-Banach Theorem and Reflexive Banach Spaces 5.9 The Banach-Schauder and Banach-Steinhaus Theorems 8.5 The Lax-Milgram Theorem We have also added a final subsection on Lebesgue integration at the end of Chap- ter 0, listing a few of the main concepts and results on Lebesgue integration in Rn. In addition, the titles of a few sections from the second edition have been changed slightly to more clearly bring out the material already contained in the sections, again to help connect the material in the sections to presentations of these topics appearing in standard references. The new section titles are: 4.4 Contractions and the Banach Fixed-Point Theorem 4.5 Hilbert Spaces and the Projection Theorem 4.7 Linear Functionals and the Riesz Representation Theorem 9.1 Introduction and Basic Fixed-Point Techniques The bibliographies at the end of the chapters in the second edition have also been updated for the third edition, but we have likely left out many outstanding new books and papers that should have been included, and we apologize in advance for all such omissions. IVAR STAKGOLD La Jolla, California November 2010 When Ivar asked me to consider joining him on a third edition of his well-known and popular book, Green’s Functions and Boundary-Value Problems , I was a bit in- timidated; not only had it been a standard reference for me for many years, but it is also used as the main text for the first-year graduate applied analysis sequence in a number of applied mathematics doctoral programs around the country. However, I soon realized it was an opportunity for me to add the material that I feel is often missing from first-year graduate courses in modern applied mathematics, namely, ad- ditional foundational material from analysis and approximation theory to support the design, development, and analysis of effective and reliable computational methods for partial differential equations. Although there are some wonderful books covering applied mathematics (such as Ivar’s) and some equally strong books on numerical analysis, the bridge between them (built with linear functional analysis, approxima- tion theory, and nonlinear analysis) is often mostly missing in these same books. There are a number of books devoted entirely to building this bridge; however, our goal for the third edition was to add just the right subset of this material so that a course based on this single book, combined with a course based on a strong graduate numerical analysis book, would provide a solid foundation for applied mathematics PREFACE TO THIRD EDITION xiii students in our mathematics doctoral program and in our interdisciplinary Computa- tional Science, Mathematics, and Engineering Graduate Program at UCSD. After spending substantial time with the second edition of the book over the last year, my appreciation for Ivar’s original book has only grown. The second edition is a unique combination of modeling, real analysis, linear functional analysis and op- erator theory, partial differential equations, integral equations, nonlinear functional analysis, and applications. The book manages to present the topics in a friendly, in- formal way, and at the same time gives the real theorems, with real proofs, when they are called for. The changes that I recommended we make to the second edition (as Ivar outlined above) were mostly to draw out the existing structure of the book, and also to add in a few results from linear functional analysis to complete the material where it was needed to support the new final chapter of the book. Since those of us who have worked closely with the second edition are very familiar with exactly where to find particular topics, one of my goals for the third edition was to preserve as much of the second edition as possible, right down to theorem, equation, and ex- ercise numbers within the sections of each chapter. This is why I have tried to fit all of the new material into new sections appearing at the end of existing chapters, and into the new final chapter appearing at the end of the book. The index to the second edition also provided finer-grained access to the book than did the table of contents; I always found this a very valuable part of the second edition, so I attempted to pre- serve the entire second edition index as a subset of the third edition index. My hope is that as a consequence of our efforts, the third edition of the book will be viewed as a useful superset of the second edition, with new material on approximation theory and methods, together with some additional supporting analysis material. The third edition contains approximately 30% new material not found in the sec- ond edition. The longest chapter is now the new final chapter (Chapter 10) on ap- proximation theory and methods. We considered splitting it into two chapters, but it seems to hold together well as a single chapter. In addition to the new material in Chapter 10, we have added material to Chapters 2, 4, 5, and 8 as Ivar outlined above. Chapter 2 now contains an early introduction to Sobolev spaces based on weak differentiation, and Chapter 8 now includes the Lax-Milgram Theorem and some related tools. Chapters 4 and 5 now provide a gentle introduction to many of the central concepts and theorems in linear functional analysis and operator theory, as needed by most first-year graduate students working in applied analysis and applied partial differential equations. Some of the new material in Chapter 10 is a bit more advanced than some of the other sections of the book; however, this material builds only on (old and new) material found in Chapters 2, 4, 5, and 8, with the support of a few new paragraphs added to the end of Chapter 0 (on Lebesgue integration). The only exception is perhaps the last example in Section 10.4, chosen from mathemat- ical physics to illustrate the combined use of several tools from nonlinear analysis and approximation theory; it requires a bit of familiarity with the notation used in differential geometry. A brief word about the numbering system used in third edition is in order, since we are departing substantially from the convention used in the previous two editions (as outlined in the preface to the first edition). The book is now divided into eleven xiv PREFACE TO THIRD EDITION chapters (beginning with Chapter 0), with the inclusion of a new final chapter (Chap- ter 10). Each chapter is divided into numbered sections, and equations are numbered by chapter, section, and equation within each section. For example, a reference to equation (8.5.2) is to the second numbered equation appearing in Section 5 of Chap- ter 8. Similarly, all definitions, theorems, corollaries, lemmas, and the like, as well as exercises, are numbered using the same convention. This convention makes the third edition easier to navigate than the first two editions, with a simple glance at a typical page revealing precisely the section and chapter in which the page appears. However, it also preserves the numbering of items from the second edition; for ex- ample, equation (5.2) of Chapter 8 in the second edition is numbered as (8.5.2) in the third edition. Note that some objects remain unnumbered if they were unnum- bered in the first two editions (for example, a theorem that is not referred to later in the book). To simplify the presentation without losing the advantages of this num- bering convention, we make three consistent exceptions: Figures are numbered only by chapter and figure within the chapter; examples and remarks are numbered only within the section; and the Bibliography continues to consist of a chapter-specific list of references immediately following the chapter, ordered alphabetically. Citations to references are now also numbered within the referring text; for example, a citation to reference [3] occurring within a chapter refers to the third reference appearing in the list of references at the end of the chapter. I would like to thank my family (Mai, Mason, and Makenna) for their patience during the last few months as I focused on the book. I would also like to thank the faculty in the Center for Computational Mathematics at UCSD, and in particular Randy Bank, Philip Gill, and Jim Bunch, for the support and encouragement they have given me over the last ten years. I am also indebted to the Center for Theoretical Biological Physics, the National Biomedical Computation Resource, the National Science Foundation, the National Institutes of Health, the Department of Energy, and the Department of Defense for their ongoing support of my research. I must express my appreciation for the interactions I have had with Randy Bank, Long Chen, Don Estep, Gabriel Nagy, Gantumur Tsogtgerel, and Jinchao Xu, as each played a role in the development of my understanding of much of the material I wrote for the book. I would also like to thank Ari Stern, Ryan Szypowski, Yunrong Zhu, and Jonny Serencsa for reading the new material carefully and catching mistakes. Finally, I am grateful to my friend and mentor Herb Keller, who greatly influenced my work over the last fifteen years, and this is reflected in the topics that I chose to include in the book. Herb was my postdoctoral advisor at Caltech from 1993 to 1997, and after retiring from Caltech around 2000, he moved to San Diego to join our research group at UCSD. We thoroughly enjoyed the years Herb was with us at the Center (attending the weekly seminars in his biking outfit, after biking down the coast from Leucadia). Unfortunately, Herb passed away just before Ivar joined our research group in 2008; otherwise, we might have had three authors on this new edition of the book. MICHAEL HOLST La Jolla, California November 2010 PREFACE TO THE SECOND EDITION The field of applied mathematics has evolved considerably in the nearly twenty years since this book’s first edition. To incorporate some of these changes, the publishers and I decided to undertake a second edition. Although many fine books on related subjects have appeared in recent years, we believe that the favorable reception ac- corded the first edition— as measured by adoptions and reviews— justifies the effort involved in a new edition. My basic purpose is still to prepare the reader to use differential and integral equa- tions to attack significant problems in the physical sciences, engineering, and applied mathematics. Throughout, I try to maintain a balance between sound mathematics and meaningful applications. The principal changes in the second edition are in the areas of modeling, Fourier analysis, fixed-point theorems, inverse problems, asymp- totics, and nonlinear methods. The exercises, quite a few of which are new, are rarely routine and occasionally can even be considered extensions of the text. Let me now turn to a chapter-by-chapter list of the major changes. Chapter 0 [Preliminaries] has assumed a more important role. It is now the start- ing point for a discussion of the relation among the four alternative formulations of physical problems: integral balance law, boundary value problem, weak form (also known as the principle of virtual work), and variational principle. I have also added new modeling examples in climatology, population dynamics, and fluid flow. xv xvi PREFACE TO SECOND EDITION Chapter 1 [Green’s functions: intuitive ideas] contains some revisions in exposi- tion, particularly in regard to continuous dependence on the data. In Chapter 2 [The theory of distributions], the treatment of Fourier analysis has been extended to include Discrete and Fast transforms, band-limited functions, and the sampling theorem using the sinc function. Chapter 3 [One-dimensional boundary value problems] now includes a more thor- ough treatment of least-squares solutions and pseudo-inverses. The ideas are intro- duced through a discussion of unbalanced systems (underdetermined or overdeter- mined). Chapter 4 has been retitled “Hilbert and Banach spaces,” reflecting an increased emphasis on normed spaces at the expense of general metric spaces. The material on contractions is rewritten from this point of view with some new examples. Chapter 5 [Operator theory] is virtually unchanged. Chapter 6 [Integral equations] now includes a treatment of Tychonov regular- ization for integral equations of the first kind, an important aspect of the study of ill-posed inverse problems. Some new examples of integral equations are presented and there is a short discussion of singular-value decomposition. Part of the material on integrodifferential equations has been deleted. Chapter 7 [Spectral theory of second-order differential operators] is basically un- changed. In Chapter 8 [Partial differential equations], I have added a more comprehensive treatment of the spectral properties of the Laplacian, including a discussion of recent results on isospectral problems. The asymptotic behavior of the heat equation is examined. A brief introduction to the finite element method is incorporated in a slightly revised section on variational principles. Chapter 9 [Nonlinear problems] contains a new subsection comparing the three major fixed-point theorems: the Schauder theorem, the contraction theorem of Chap- ter 4, and the theorem for order-preserving maps, which is used extensively in the remainder of Chapter 9. I have also included a study of the phenomena of finite-time extinction and blow-up for nonlinear reaction-diffusion problems. There now remains the pleasant task of acknowledging my debt to the students and teachers who commented on the first edition and diplomatically muted their criticism! I am particularly grateful to my friends Stuart Antman of the University of Maryland, W. Edward Olmstead of Northwestern University, and David Colton and M. Zuhair Nashed of the University of Delaware, who generously provided me with ideas and encouragement. The new material in Chapter 9 owes much to my overseas collaborators, Catherine Bandle (University of Basel) and J. Ildefonso Diaz (Universidad Complutense, Madrid). The TEX preparation of the manuscript was in the highly skilled hands of Linda Kelly and Pamela Haverland. IVAR STAKGOLD Newark, Delaware September 1997 PREFACE TO THE FIRST EDITION As a result of graduate-level adoptions of my earlier two-volume book, Boundary Value Problems of Mathematical Physics , I received many constructive suggestions from users. One frequent recommendation was to consolidate and reorganize the topics into a single volume that could be covered in a one-year course. Another was to place additional emphasis on modeling and to choose examples from a wider variety of physical applications, particularly some emerging ones. In the meantime my own research interests had turned to nonlinear problems, so that, inescapably, some of these would also have to be included in any revision. The only way to incorporate these changes, as well as others, was to write a new book, whose main thrust, however, remains the systematic analysis of boundary value problems. Of course some topics had to be dropped and others curtailed, but I can only hope that your favorite ones are not among them. My book is aimed at graduate students in the physical sciences, engineering, and applied mathematics who have taken the typical “methods” course that includes vec- tor analysis, elementary complex variables, and an introduction to Fourier series and boundary value problems. Why go beyond this? A glance at modern publications in science and engineering provides the answer. To the lament of some and the delight of others, much of this literature is deeply mathematical. I am referring not only to areas such as mechanics and electromagnetic theory that are traditionally mathemati- cal but also to relative newcomers to mathematization, such as chemical engineering, xvii xviii PREFACE TO FIRST EDITION materials science, soil mechanics, environmental engineering, biomedical engineer- ing, and nuclear engineering. These fields give rise to challenging mathematical problems whose flavor can be sensed from the following short list of examples; in- tegrodifferential equations of neutron transport theory, combined diffusion and re- action in chemical and environmental engineering, phase transitions in metallurgy, free boundary problems for dams in soil mechanics, propagation of impulses along nerves in biology. It would be irresponsible and foolish to claim that readers of my book will become instantaneous experts in these fields, but they will be prepared to tackle many of the mathematical aspects of the relevant literature. Next, let me say a few words about the numbering system. The book is divided into ten chapters, and each chapter is divided into sections. Equations do notcarry a chapter designation. A reference to, say, equation 4.32 is to the thirty-second num- bered equation in Section 4 of the chapter you happen to be reading. The same system is used for figures and exercises, the latter being found at the end of sections. The exercises, by the way, are rarely routine and, on occasion, contain substantial extensions of the main text. Examples do not carry any section designation and are numbered consecutively within a section, even though there may be separate clus- ters of examples within the same section. Some theorems have numbers and others do not; those that do are numbered in a sequence within a section— Theorem 1, Theorem 2, and so on. A brief description of the book’s contents follows. No attempt is made to mention all topics covered; only the general thread of the development is indicated. Chapter 0 presents background material that consists principally of careful deriva- tions of several of the equations of mathematical physics. Among them are the equa- tions of heat conduction, of neutron transport, and of vibrations of rods. In the last- named derivation an effort is made to show how the usual linear equations for beams and strings can be regarded as first approximations to nonlinear problems. There are also two short sections on modes of convergence and on Lebesgue integration. Many of the principal ideas related to boundary value problems are introduced on an intuitive level in Chapter 1. A boundary value problem (BVP, for short) consists of a differential equation Lu=fwith boundary conditions of the form Bu=h. The pair (f,h) is known collectively as the data for the problem, and uis the response to be determined. Green’s function is the response when frepresents a concentrated unit source and h= 0. In terms of Green’s function, the BVP with arbitrary data can be solved in a form that shows clearly the dependence of the solution on the data. Various examples are given, including some multidimensional ones, some in- volving interface conditions, and some initial value problems. The useful notion of a well-posed problem is discussed, and a first look is taken at maximum principles for differential equations. Chapter 2 deals with the theory of distributions, which provides a rigorous math- ematical framework for singular sources such as the point charges, dipoles, line charges, and surface layers of electrostatics. The notion of response to such sources is made precise by defining the distributional solution of a differential equation. The related concepts of weak solution, adjoint, and fundamental solution are also in- PREFACE TO FIRST EDITION xix troduced. Fourier series and Fourier transforms are presented in both classical and distributional settings. Chapter 3 returns to a more detailed study of one-dimensional linear boundary value problems. To an equation of order pthere are usually associated pindependent boundary conditions involving derivatives of order less than pat the endpoints aand bof a bounded interval. If the corresponding BVP with 0 data has only the trivial solution, then the BVP with arbitrary data has one and only one solution which can be expressed in terms of Green’s function. If, however, the BVP with 0 data has a nontrivial solution, certain solvability conditions must be satisfied for the BVP with arbitrary data to have a solution. These statements are formulated precisely in an alternative theorem, which recurs throughout the book in various forms. When the BVP with 0 data has a nontrivial solution, Green’s function cannot be constructed in the ordinary way, but some of its properties can be salvaged by using a modified Green’s function, defined in Section 5. Chapter 4 begins the study of Hilbert spaces. A Hilbert space is the proper set- ting for many of the linear problems of applied analysis. Though its elements may be functions or abstract “vectors,” a Hilbert space enjoys all the algebraic and ge- ometric properties of ordinary Euclidean space. A Hilbert space is a linear space equipped with an inner product that induces a natural notion of distance between elements, thereby converting it into a metric space which is required to be com- plete. Some of the important geometric properties of Hilbert spaces are developed, including the projection theorem and the existence of orthonormal bases for sepa- rable spaces. Metric spaces can be useful quite apart from any linear structure. A contraction is a transformation on a metric space that uniformly reduces distances between pairs of points. A contraction on a complete metric space has a unique fixed point that can be calculated by iteration from any initial approximation. Examples demonstrate how to use these ideas to prove uniqueness and constructive existence for certain classes of nonlinear differential equations and integral equations. Chapter 5 examines the theory of linear operators on a separable Hilbert space, particularly integral and differential operators, the latter being unbounded operators. The principal problem of operator theory is the solution of the equation Au=f, whereAis a linear operator and fan element of the space. A thorough discussion of this problem leads again to adjoint operators, solvability conditions, and alternative theorems. Additional insight is obtained by considering the inversion of the equation Au−λu=f, which leads to the idea of the spectrum, a generalization of the more familiar concept of eigenvalue. For compact operators (which include most integral operators) the inversion problem is essentially solved by the Riesz-Schauder theory of Section 7. Section 8 relates the spectrum of symmetric operators to extremal principles for the Rayleigh quotient. Throughout, the theory is illustrated by specific examples. In Chapter 6 the general ideas of operator theory are specialized to integral equa- tions. Integral equations are particularly important as alternative formulations of boundary value problems. Special emphasis is given to Fredholm equations with symmetric Hilbert-Schmidt kernels. For the corresponding class of operators, the nonzero eigenvalues and associated eigenfunctions can be characterized through suc- xx PREFACE TO FIRST EDITION cessive extremal principles, and it is then possible to give a complete treatment of the inhomogeneous equation. The last section discusses the Ritz procedure for es- timating eigenvalues, as well as other approximation methods for eigenvalues and eigenfunctions. There is also a brief introduction to integrodifferential operators in Exercises 5.3 to 5.8. Chapter 7 extends the Sturm-Liouville theory of second-order ordinary differen- tial equations to the case of singular endpoints. It is shown, beginning with the regular case, how the necessarily discrete spectrum can be constructed from Green’s function. A formal extension of this relationship to the singular case makes it pos- sible to calculate the spectrum, which may now be partly continuous. The transi- tion from regular to singular is analyzed rigorously for equations of the first order, but the Weyl classification for second-order equations is given without proof. The eigenfunction expansion in the singular case can lead to integral transforms such as Fourier, Hankel, Mellin, and Weber. It is shown how to use these transforms and their inversion formulas to solve partial differential equations in particular geometries by separation of variables. Although partial differential equations have appeared frequently as examples in earlier chapters, they are treated more systematically in Chapter 8. Examination of the Cauchy problem— the appropriate generalization of the initial value problem to higher dimensions— gives rise to a natural classification of partial differential equa- tions into hyperbolic, parabolic, and elliptic types. The theory of characteristics for hyperbolic equations is introduced and applied to simple linear and nonlinear exam- ples. In the second and third sections various methods (Green’s functions, Laplace transforms, images, etc.) are used to solve BVPs for the wave equation, the heat equation, and Laplace’s equation. The simple and double layers of potential theory make it possible to reduce the Dirichlet problem to an integral equation on the bound- ary of the domain, thereby providing a rather weak existence proof. In Section 4 a stronger existence proof is given, using variational principles. Two-sided bounds for some functionals of physical interest, such as capacity and torsional rigidity, are obtained by introducing complementary principles. Another application involving level-line analysis is also given, and there is a very brief treatment of unilateral con- straints and variational inequalities. Finally, in Chapter 9, a number of methods applicable to nonlinear problems are developed. Section 1 points out some of the features that distinguish nonlinear prob- lems from linear ones and illustrates these differences through some simple exam- ples. In Section 2 the principal qualitative results of branching theory (also known as bifurcation theory) are presented. The phenomenon of bifurcation is understood most easily in terms of the buckling of a rod under compressive thrust. As the thrust is increased beyond a certain critical value, the state of simple compression gives way to the buckled state with its appreciable transverse deflection. Section 3 shows how a variety of linear problems can be handled by perturbation theory (inhomoge- neous problems, eigenvalue problems, change in boundary conditions, domain per- turbations). These techniques, as well as monotone methods, are then adapted to the solution of nonlinear BVPs. The concluding section discusses the possible loss of stability of the basic steady state when an underlying parameter is allowed to vary. PREFACE TO FIRST EDITION xxi I have already acknowledged my debt to the students and teachers who were kind enough to comment on my earlier book. There are, however, two colleagues to whom I am particularly grateful: Stuart Antman, who generously contributed the ideas underlying the derivation of the equations for rods in Chapter 0, and W. Edward Olmstead, who suggested some of the examples on contractions in Chapter 4 and on branching in Chapter 9. IVAR STAKGOLD Newark, Delaware September 1979