Potential Theory

Potential Theory


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.

Table of contents

Front Matter....Pages i-xiv
Preliminaries....Pages 1-8
Laplaces Equation....Pages 9-62
The Dirichlet Problem....Pages 63-120
Green Functions....Pages 121-164
Negligible Sets....Pages 165-213
Dirichlet Problem for Unbounded Regions....Pages 215-258
Energy....Pages 259-284
Interpolation and Monotonicity....Pages 285-320
Newtonian Potential....Pages 321-351
Elliptic Operators....Pages 353-390
Apriori Bounds....Pages 391-411
Oblique Derivative Problem....Pages 413-451
Application to Diffusion Processes....Pages 453-477
Back Matter....Pages 479-485


  • Author: Lester L. Helms (auth.)
  • Edition: 2
  • Publication Date: 2014
  • Publisher: Springer-Verlag London
  • ISBN-13: 9781447164210, 9781447164227
  • Pages: 494
  • Format: pdf
  • Size: 4.6M
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