Harnack's inequality for degenerate and singular parabolic equations

Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri

Degenerate and singular parabolic equations have been the subject of extensive research for the last 25 years. Despite important achievements, the issue of the Harnack inequality for non-negative solutions to these equations, both of p-Laplacian and porous medium type, while raised by several authors, has remained basically open. Recently considerable progress has been made on this issue, to the point that, except for the singular sub-critical range, both for the p-laplacian and the porous medium equations, the theory is reasonably complete. It seemed therefore timely to trace a comprehensive overview, that would highlight the main issues and also the problems that still remain open. The authors give a comprehensive treatment of the Harnack inequality for non-negative solutions to p-laplace and porous medium type equations, both in the degenerate (p2 or m1) and in the singular range (1p<2 or 0m<1), starting from the notion of solution and building all the necessary technical tools. The book is self-contained. Building on a similar monograph by the first author, the authors of the present book focus entirely on the Harnack estimates and on their applications: indeed a number of known regularity results are given a new proof, based on the Harnack inequality. It is addressed to all professionals active in the field, and also to advanced graduate students, interested in understanding the main issues of this fascinating research field.

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  • Preface.- 1. Introduction.- 2. Preliminaries.- 3. Degenerate and Singular Parabolic Equations.- 4. Expansion of Positivity.- 5. The Harnack Inequality for Degenerate Equations.- 6. The Harnack Inequality for Singular Equations.- 7. Homogeneous Monotone Singular Equations.- Appendix A.- Appendix B.- Appendix C.- References.- Index.

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書名 Harnack's inequality for degenerate and singular parabolic equations
著作者等 DiBenedetto, Emmanuele
Vespri Vincenzo
Gianazza Ugo
シリーズ名 Springer monographs in mathematics
出版元 Springer
刊行年月 c2012
版表示 2012
ページ数 xiii, 278 p.
大きさ 25 cm
ISBN 9781461415831
NCID BB07696287
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言語 英語
出版国 アメリカ合衆国