An important part of the theory of partial differential equations is the theory of boundary problems for elliptic equations and systems of equations. Among such problems those of greatest interest are the so-called non-Fredholm boundary prob lems, whose investigation reduces, as a rule, to the study of singular integral equa tions, where the Fredholm alternative is violated for these problems. Thanks to de velopments in the theory of one-dimensional singular integral equations [28, 29], boundary problems for elliptic equations with two independent variables have been completely studied at the present time [13, 29], which cannot be said about bound ary problems for elliptic equations with many independent variables. A number of important questions in this area have not yet been solved, since one does not have sufficiently general methods for investigating them. Among the boundary problems of great interest is the oblique derivative problem for harmonic functions, which can be formulated as follows: In a domain D with sufficiently smooth boundary r find a harmonic function u(X) which, on r, satisfies the condition n n ~ au ~ . . . :;. . ai (X) ax. = f (X), . . . :;. . [ai (X)]2 = 1, i=l t i=l where aI, . . . , an,fare sufficiently smooth functions defined on r. Obviously the left side of the boundary condition is the derivative of the function u(X) in the direction of the vector P(X) with components al (X), . . . , an(X).

An important part of the theory of partial differential equations is the theory of boundary problems for elliptic equations and systems of equations. Among such problems those of greatest interest are the so-called non-Fredholm boundary prob lems, whose investigation reduces, as a rule, to the study of singular integral equa tions, where the Fredholm alternative is violated for these problems. Thanks to de velopments in the theory of one-dimensional singular integral equations [28, 29], boundary problems for elliptic equations with two independent variables have been completely studied at the present time [13, 29], which cannot be said about bound ary problems for elliptic equations with many independent variables. A number of important questions in this area have not yet been solved, since one does not have sufficiently general methods for investigating them. Among the boundary problems of great interest is the oblique derivative problem for harmonic functions, which can be formulated as follows: In a domain D with sufficiently smooth boundary r find a harmonic function u(X) which, on r, satisfies the condition n n ~ au ~ . . . :;. . ai (X) ax. = f (X), . . . :;. . [ai (X)]2 = 1, i=l t i=l where aI, . . . , an,fare sufficiently smooth functions defined on r. Obviously the left side of the boundary condition is the derivative of the function u(X) in the direction of the vector P(X) with components al (X), . . . , an(X).

An important part of the theory of partial differential equations is the theory of boundary problems for elliptic equations and systems of equations. Among such problems those of greatest interest are the so-called non-Fredholm boundary prob lems, whose investigation reduces, as a rule, to the study of singular integral equa tions, where the Fredholm alternative is violated for these problems. Thanks to de velopments in the theory of one-dimensional singular integral equations [28, 29], boundary problems for elliptic equations with two independent variables have been completely studied at the present time [13, 29], which cannot be said about bound ary problems for elliptic equations with many independent variables. A number of important questions in this area have not yet been solved, since one does not have sufficiently general methods for investigating them. Among the boundary problems of great interest is the oblique derivative problem for harmonic functions, which can be formulated as follows: In a domain D with sufficiently smooth boundary r find a harmonic function u(X) which, on r, satisfies the condition n n ~ au ~ . . . :;. . ai (X) ax. = f (X), . . . :;. . [ai (X)]2 = 1, i=l t i=l where aI, . . . , an,fare sufficiently smooth functions defined on r. Obviously the left side of the boundary condition is the derivative of the function u(X) in the direction of the vector P(X) with components al (X), . . . , an(X).

The ?rst six chapters of this book are revised versions of the same chapters in the author’s 1969 book, Introduction to Potential Theory. Atthetimeof the writing of that book, I had access to excellent articles,books, and lecture notes by M. Brelot. The clarity of these works made the task of collating them into a single body much easier. Unfortunately, there is not a similar collection relevant to more recent developments in potential theory. A n- comer to the subject will ?nd the journal literature to be a maze of excellent papers and papers that never should have been published as presented. In the Opinion Column of the August, 2008, issue of the Notices of the Am- ican Mathematical Society, M. Nathanson of Lehman College (CUNY) and (CUNY) Graduate Center said it best “. . . When I read a journal article, I often ?nd mistakes. Whether I can ?x them is irrelevant. The literature is unreliable. ” From time to time, someone must try to ?nd a path through the maze. In planning this book, it became apparent that a de?ciency in the 1969 book would have to be corrected to include a discussion of the Neumann problem, not only in preparation for a discussion of the oblique derivative boundary value problem but also to improve the basic part of the subject matter for the end users, engineers, physicists, etc.

The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose: first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals; and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, in order that the book may present sound ideals to the student, and also serve the mathematician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem (Gauss', or Green's theorem) on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Dirichlet problem.

Potential Theory presents a clear path from calculus to classical potential theory and beyond, with the aim of moving the reader into the area of mathematical research as quickly as possible. The subject matter is developed from first principles using only calculus. Commencing with the inverse square law for gravitational and electromagnetic forces and the divergence theorem, the author develops methods for constructing solutions of Laplace's equation on a region with prescribed values on the boundary of the region. The latter half of the book addresses more advanced material aimed at those with the background of a senior undergraduate or beginning graduate course in real analysis. Starting with solutions of the Dirichlet problem subject to mixed boundary conditions on the simplest of regions, methods of morphing such solutions onto solutions of Poisson's equation on more general regions are developed using diffeomorphisms and the Perron-Wiener-Brelot method, culminating in application to Brownian motion. In this new edition, many exercises have been added to reconnect the subject matter to the physical sciences. This book will undoubtedly be useful to graduate students and researchers in mathematics, physics and engineering.

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