Introduction to Heat Potential Theory (Mathematical Surveys and Monographs)

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. 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. ... 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.--publisher's Description. The Heat Operator, Temperatures And Mean Values -- The Poisson Integral For A Circular Cylinder -- Subtemperatures And The Dirichlet Problem On Convex Domains Of Revolution -- Temperatures On An Infinite Strip -- Classes Of Subtemperatures On An Infinite Strip -- Green Functions And Heat Potentials -- Polar Sets And Thermal Capacity -- The Dirichlet Problem On Arbitrary Open Sets -- The Thermal Fine Topology. Neil A. Watson. Includes Bibliographical References (p. 259-262) And Index. Cover 1 Title page 2 Contents 4 Preface 8 Notation and terminology 12 The heat operator, temperatures and mean values 16 The Poisson integral for a circular cylinder 50 Subtemperatures and the Dirichlet problem on convex domains of revolution 68 Temperatures on an infinite strip 100 Classes of subtemperatures on an infinite strip 120 Green functions and heat potentials 142 Polar sets and thermal capacity 174 The Dirichlet problem on arbitrary open sets 210 The thermal fine topology 246 Bibliography 274 Index 278 Back Cover 282 This is the first book 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.