Potential Theory

sheldon axler at all;

The first six chapters of this book are revised versions of the same chapters in the author’s 1969 book, Introduction to Potential Theory. At the time of 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 newcomer to the subject will find 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 American Mathematical Society, M. Nathanson of Lehman College (CUNY) and (CUNY) Graduate Center said it best “...When I read a journal article, I often find mistakes. Whether I can fix them is irrelevant. The literature is unreliable.” From time to time, someone must try to find a path through the maze.
In planning this book, it became apparent that a deficiency 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. Generally speaking, the choice of topics was intended to make the book more pragmatic and less esoteric while exposing the reader to the major accomplishments by some of the most prominent mathematicians of the eighteenth through twentieth centuries. Most of these accomplishments had their origin in practical matters as, for example, Green’s assumption that there is a function, which he called a potential function, which could be used to solve problems related to electromagnetic fields.
This book is targeted primarily at students with a background in a senior or graduate level course in real analysis which includes basic material on topology, measure theory, and Banach spaces. I have tried to present a clear path from the calculus to classic potential theory and then to recent work on elliptic partial differential equations using potential theory methods. The goal has been to move the reader into a fertile area of mathematical research as quickly as possible.
The author is indebted to L. H¨ormander and K. Miller for their prompt responses to queries about the details of some of their proofs. The author also thanks Karen Borthwick of Springer, UK, for guiding the author through the publication process.