Bergman S., Schiffer M. Kernel Functions and Elliptic Differential Equations in Mathematical Physics

Academic Press, 1953. — 432 p.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.