**Mathematical Surveys and Monographs**

Volume: 182;
2012;
266 pp;
Hardcover

MSC: Primary 31; 35;

Print ISBN: 978-0-8218-4998-9

Product Code: SURV/182

List Price: $86.00

Individual Member Price: $68.80

**Electronic ISBN: 978-0-8218-8792-9
Product Code: SURV/182.E**

List Price: $86.00

Individual Member Price: $68.80

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#### Supplemental Materials

# Introduction to Heat Potential Theory

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*Neil A. Watson*

This book is the first to be devoted entirely to the potential theory of the heat equation, and thus deals with time dependent potential theory. Its purpose is to give a logical, mathematically precise introduction to a subject where previously many proofs were not written in detail, due to their similarity with those of the potential theory of Laplace's equation.

The approach to subtemperatures is a recent one, based on the Poisson integral representation of temperatures on a circular cylinder. Characterizations of subtemperatures in terms of heat balls and modified heat balls are proved, and thermal capacity is studied in detail. The generalized Dirichlet problem on arbitrary open sets is given a treatment that reflects its distinctive nature for an equation of parabolic type. Also included is some new material on caloric measure for arbitrary open sets.

Each chapter concludes with bibliographical notes and open questions. The reader should have a good background in the calculus of functions of several variables, in the limiting processes and inequalities of analysis, in measure theory, and in general topology for Chapter 9.

#### Table of Contents

# Table of Contents

## Introduction to Heat Potential Theory

- Cover Cover11 free
- Title page i2 free
- Contents iii4 free
- Preface vii8 free
- Notation and terminology xi12 free
- The heat operator, temperatures and mean values 116 free
- The Poisson integral for a circular cylinder 3550
- Subtemperatures and the Dirichlet problem on convex domains of revolution 5368
- Temperatures on an infinite strip 85100
- Classes of subtemperatures on an infinite strip 105120
- Green functions and heat potentials 127142
- Polar sets and thermal capacity 159174
- The Dirichlet problem on arbitrary open sets 195210
- The thermal fine topology 231246
- Bibliography 259274
- Index 263278 free
- Back Cover Back Cover1282

#### Readership

Graduate students and research mathematicians interested in partial differential equations and potential theory.