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Potential Theory

  • Textbook
  • © 2009

Overview

Editors:
  1. Lester L. Helms

    1. Department of Mathematics, University of Illinois, Urbana, USA

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  • Written by the author of Introduction to Potential Theory, this is a new and modern textbook that introduces all the important concepts of classical potential theory
  • Equips readers for further study in elliptic partial differential equations, axiomatic potential theory, and the interplay between probability theory and potential theory
  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xi
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  2. Preliminaries

    • Lester L. Helms
    Pages 1-6

  3. Laplace's Equation

    • Lester L. Helms
    Pages 7-52

  4. The Dirichlet Problem

    • Lester L. Helms
    Pages 53-105

  5. Green Functions

    • Lester L. Helms
    Pages 107-147

  6. Negligible Sets

    • Lester L. Helms
    Pages 149-196

  7. Dirichlet Problem for Unbounded Regions

    • Lester L. Helms
    Pages 197-240

  8. Energy

    • Lester L. Helms
    Pages 241-265

  9. Interpolation and Monotonicity

    • Lester L. Helms
    Pages 267-301

  10. Newtonian Potential

    • Lester L. Helms
    Pages 303-331

  11. Elliptic Operators

    • Lester L. Helms
    Pages 333-369

  12. Apriori Bounds

    • Lester L. Helms
    Pages 371-389

  13. Oblique Derivative Problem

    • Lester L. Helms
    Pages 391-429

  14. Back Matter

    Pages 431-441
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Keywords

About this book

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.

Reviews

From the reviews:

“The author sets the goal of the book as getting the reader from real analysis to the front line of potential theory as quickly as possible. … Each chapter begins with some physical and historical context … . The excellent index is also very helpful in navigating the material. … a fine text for self-study, reference, or a graduate course. Researchers in the field will consider it a standard and those in an adjacent field … will also find it a valuable reference.” (Bill Wood, The Mathematical Association of America, May, 2010)

“The first part of the book deals with the basics of classical potential theory while the rest of the book deals with the solution to elliptic partial differential equations with various boundary conditions. … Proofs are given in easy to follow detail. … the book is very suitable as a textbook … . On the whole, this book is a very useful addition to available resources. … There is also an index, a notation guide and an extensive bibliography.” (P. Lappan, Mathematical Reviews, Issue 2011 a)

Editors and Affiliations

  • Department of Mathematics, University of Illinois, Urbana, USA

    Lester L. Helms

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