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Kernel Functions and Elliptic Differential Equations in Mathematical Physics (Dover Books on Mathematics): Stefan Bergman, Menahem Schiffer

The subject of this book is the theory of boundary value problems in partial differential equations. This theory plays a central role in various fields of pure and applied mathematics, theoretical physics, and engineering, and has already been dealt with in numerous books and articles. This book discusses a portion of the theory from a unifying point of view. The solution of a partial differential equation of elliptic type is a functional of the boundary values, the coefficients of the differential equation, and the domain considered. The dependence of the solution upon its boundary values has been studied extensively, but its dependence upon the coefficients and upon the domain is almost as important. The problem of the variation of the solution with that of the coefficients of the equation is closely related to questions of stability, which are of decisive importance in many applications. The knowledge of how the solution of a differential equation varies with a change of coefficients or domain permits us to concentrate on the study of simple equations in simple domains and to derive qualitative results from them.

When studying the relationship of a solution to the boundary values, one is led to introduce certain fundamental solutions: Green's, Neumann's, and Robin's functions. Every solution can be expressed in terms of one of these functions, and it is therefore natural to emphasize a systematic study of them. In this way, we deal only with a few well-defined functions and their interrelations and obtain a clear insight into the structure of all possible solutions of the differential equation. The fundamental solutions depend upon two argument points, are symmetric in both, and are a function of each separately. Solutions of this type are called kernels, and after linear operations extended over one variable they still represent solutions of the equation in the other variable. The systematic treatment of the various kernels and their properties is the main object of this book. In the treatment of the fundamental solutions, certain combinations play a particularly important role. Although Green's, Neumann's, and Robin's functions possess singular points in the domain of definition, combinations of them can be found which are regular throughout the whole domain and which are, therefore, particularly amenable to theoretical and numerical treatment. For example, the difference between any two fundamental solutions is a regular kernel and possesses various important properties with respect to its boundary behavior.

The book consists of two main parts. The first part gives a survey of boundary value problems occurring in some branches of theoretical physics. The various fundamental solutions are introduced in a heuristic way, and their physical significance is studied. Various concepts can be unified by concentrating upon these particular kernels. The common mathematical background of so widely varying theories as heat conduction, hydrodynamics, electrostatics, magnetostatics, and elasticity is shown. In addition to its own intrinsic interest, the material of this part provides illustrations and adds significance to the second part of the book, which is devoted to the exact mathematical formulation of problems and the methods involved. For the sake of simplicity we have restricted ourselves in the second part to a rather special type of partial differential equation. On the other hand, we have dealt with this equation in the greatest detail and are confident that a careful reader will be able to make applications and generalizations to similar problems which may be of interest to him.

The present book is not to be considered a textbook on partial differential equations. It assumes a fair acquaintance with the standard methods of analysis provided, for example, by the excellent books of Courant-Hilbert, FrankMises, and Jeffreys. For this reason, an engineer or a physicist with the conventional mathematical training may possibly find some parts difficult to read. We have attempted to incorporate material which is of interest and of use to this class of readers and have tried to provide a systematic and self-contained introduction to each branch of the applications treated. On the mathematical side we have included much material which has been obtained in the researches of the last few years and which we hope may lead to further research and progress in the field of partial differential equations.

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