Catalogue


Nonlinear potential theory of degenerate elliptic equations /
Juha Heinonen, Tero Kilpeläinen, Olli Martio.
imprint
Oxford ; New York : Clarendon Press ; New York : Oxford University Press, 1993.
description
v, 363 p.
ISBN
0198536690 :
format(s)
Book
Holdings
More Details
imprint
Oxford ; New York : Clarendon Press ; New York : Oxford University Press, 1993.
isbn
0198536690 :
catalogue key
3011610
 
Includes bibliographical references and index.
A Look Inside
Reviews
Review Quotes
'excellent book'S.K. Vodopyanov, Mathematical Reviews, Issue 94e
'In particular, the theory of the (degenerate) eliptic equations . . . is very readable and very comprehensively presented. . . . This is a highly recommendable, solid introduction to the nonlinear potential theory, compulsory reading for everyone who would like to work in this field.'Deutschen Mathematiker-Vereinigung
'The book is warmly recommended to specialists in potential theory, elliptic PDE's, Sobolev spaces and quasiconformal mappings.'European Mathematical Society Newsletter, No. 11, 1994
To find out how to look for other reviews, please see our guides to finding book reviews in the Sciences or Social Sciences and Humanities.
Summaries
Long Description
This book provides a detailed introduction to nonlinear potential theory based on supersolutions to certain degenerate elliptic equations of the p-Laplacian type. Recent research has shown that classical notions such as blayage, polar sets, Perron's method, and fine topology have their proper analogues in a nonlinear setting, and this book presents a coherent exposition of this natural extension of classical potential theory. Yet fundamental differences to classical potential theory exist, and in many places a new approach is mandatory. Sometimes new or long-forgotten methods emerge that are applicable to problems in classical potential theory. Quasiregular mappings constitute a natural field of applications, and a careful study of the potential theoretical aspects of these mappings is included. The principle aim of the book is to explore the ground where partial differential equations, harmonic analysis, and function theory meet. The quasilinear equations considered in this book involve a degeneracy condition given in terms of a weight function and therefore most results appear here for the first time in print. The reader interested exclusively in the unweighted theory will find new results, new proofs, and a reorganization of the material as compared to the existing literature. The book is intended for researchers and graduate students in potential theory, variational calculus, partial differential equations, and quasiconformal mappings.
Main Description
This book provides a detailed introduction to the nonlinear potential theory based on supersolutions to certain degenerate elliptic equations of the p-Laplacian type. recent research has shown that such classical notions as blayage, polar sets, Perron's method, and fine topology have theirproper analogues in a nonlinear setting, and a coherent exposition of this natural extension of classical potential theory is presented. Yet fundamental differences to classical potential theory exist, and in many places a new approach is mandatory. Sometimes new or long-forgotten methods emergethat are applicable also to problems in classical potential theory. Quasiregular mappings constitute a natural field of applications, and a careful study of the potential theoretical aspects of these mappings is included. The aim of the book is to explore the ground where partial differentialequations, harmonic analysis, and function theory meet. The quasilinear equations considered in this book involve a degeneracy condition given in terms of a weight function and therefore most results appear here for the first time in print. However, the reader interested exclusively in theunweighted theory will find new results, new proofs, and a reorganization of the material as compared to the existing literature.
Main Description
This text offers a detailed development of the necessary background for its survey of the nonlinear potential theory of superharmonic functions. Topics include the theory of weighted variational capacity; solutions and supersolutions of equation; balayage; Perron's method, barriers, and resolutivity; polar sets; harmonic measure; fine topology; more. 1993 edition.
Table of Contents
Introduction
Weighted Sobolev spaces
Capacity
Supersolutions and the obstacle problem
Refined Sobolev spaces
Variational integrals
A-harmonic functions
A superharmonic functions
Balayage
Perron's method, barriers, and resolutivity
Polar sets
A-harmonic measure
Fine topology
Harmonic morphisms
Quasiregular mappings
Ap-weights and Jacobians of quasiconformal mappings
Axiomatic nonlinear potential theory
Appendix I: The existence of solutions
Appendix II: The John-Nirenberg lemma
Bibliography
List of symbols
Index
Table of Contents provided by Publisher. All Rights Reserved.

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