Nonlinear Potential Theory of Degenerate Elliptic Equations (Dover Books on Mathematics) - Softcover

Heinonen, Juha; Kilpeläinen, Tero; Martio, Olli; Mathematics

Softcover

ISBN 10: 0486450503 ISBN 13: 9780486450506

Publisher: Dover Publications, 2006

Synopsis

A self-contained treatment appropriate for advanced undergraduates and graduate students, this text offers a detailed development of the necessary background for its survey of the nonlinear potential theory of superharmonic functions.
Starting with the theory of weighted Sobolev spaces, this treatment advances to the theory of weighted variational capacity. Succeeding chapters investigate solutions and supersolutions of equations, with emphasis on refined Sobolev spaces, variational integrals, and harmonic functions. Chapter 7 defines superharmonic functions via the comparison principle, and chapters 8 through 14 form the core of the nonlinear potential theory of superharmonic functions. Topics include balayage; Perron's method, barriers, and resolutivity; polar sets; harmonic measure; fine topology; harmonic morphisms; and quasiregular mappings. The text concludes with explorations of axiomatic nonlinear potential theory and helpful appendixes.

About this title

About the Author

Juha Heinonen (1960 2007) was Professor of Mathematics at the University of Michigan. His principal areas of research interest included quasiconformal mappings, nonlinear potential theory, and analysis on metric spaces. He was the author of over 60 research articles, including several posthumously, and two textbooks. A member of the Finnish Academy of Science and Letters, Heinonen received the Excellence in Research Award from the University of Michigan in 1997 and gave an invited lecture at the International Congress of Mathematicians in Beijing in 2002.

Review

'excellent book'
S.K. Vodopyanov, Mathematical Reviews, Issue 94e

`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

`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

Edition details

• Publisher: Dover Publications
• Publication date: 2006
• Language: English
• ISBN 10: 0486450503
• ISBN 13: 9780486450506
• Binding: Paperback
• Number of pages: 416