Hardcover ISBN:  9780821849989 
Product Code:  SURV/182 
List Price:  $92.00 
MAA Member Price:  $82.80 
AMS Member Price:  $73.60 
Electronic ISBN:  9780821887929 
Product Code:  SURV/182.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $68.80 

Book DetailsMathematical Surveys and MonographsVolume: 182; 2012; 266 ppMSC: Primary 31; 35;
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.ReadershipGraduate students and research mathematicians interested in partial differential equations and potential theory.

Table of Contents

Chapters

1. The heat operator, temperatures and mean values

2. The Poisson integral for a circular cylinder

3. Subtemperatures and the Dirichlet problem on convex domains of revolution

4. Temperatures on an infinite strip

5. Classes of subtemperatures on an infinite strip

6. Green functions and heat potentials

7. Polar sets and thermal capacity

8. The Dirichlet problem on arbitrary open sets

9. The thermal fine topology


Additional Material

Request Review Copy

Get Permissions
 Book Details
 Table of Contents
 Additional Material

 Request Review Copy
 Get Permissions
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.
Graduate students and research mathematicians interested in partial differential equations and potential theory.

Chapters

1. The heat operator, temperatures and mean values

2. The Poisson integral for a circular cylinder

3. Subtemperatures and the Dirichlet problem on convex domains of revolution

4. Temperatures on an infinite strip

5. Classes of subtemperatures on an infinite strip

6. Green functions and heat potentials

7. Polar sets and thermal capacity

8. The Dirichlet problem on arbitrary open sets

9. The thermal fine topology