Authors:
- Topic is related to nonlinear parabolic PDEs which has been an active area of research in the last decades
- Main focus includes recent breakthroughs extending known results for the prototype equations to cover a whole class of nonlinear problems in a full scale of parameter values
- Focuses on important and interesting developments in nonlinear partial differential equations
- Includes energy estimates, expansion of positivity, intrinsic scaling and very elegant measure-theoretic results
- Techniques are very general and flexible and they can be applied in many different contexts
- Includes supplementary material: sn.pub/extras
Part of the book series: Springer Monographs in Mathematics (SMM)
Buy it now
Buying options
Tax calculation will be finalised at checkout
Other ways to access
This is a preview of subscription content, log in via an institution to check for access.
Table of contents (10 chapters)
-
Front Matter
-
Back Matter
About this book
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 (p>2 or m>1) and in the singular range (1<p<2 or 0<m<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.Reviews
From the reviews:
“Degenerate and singular parabolic equations have been the subject of extensive research for the last 25 years, but the issue of the Harnack inequality has remained basically open. In the Introduction to this monograph, the authors present the history of the subject beginning with Harnack’s inequality for nonnegative harmonic functions … . The book is self-contained and 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.” (Boris V. Loginov, Zentralblatt MATH, Vol. 1237, 2012)Authors and Affiliations
-
Department of Mathematics, Vanderbilt University, Nashville, USA
Emmanuele DiBenedetto
-
, Dipt. di Matematica "F. Casorati", Universita' degli Studi di Pavia, Pavia, Italy
Ugo Gianazza
-
, Dipartimento di Matematica, Università degli studi di Firenze, Firenze, Italy
Vincenzo Vespri
Bibliographic Information
Book Title: Harnack's Inequality for Degenerate and Singular Parabolic Equations
Authors: Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
Series Title: Springer Monographs in Mathematics
DOI: https://doi.org/10.1007/978-1-4614-1584-8
Publisher: Springer New York, NY
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Science+Business Media, LLC 2012
Hardcover ISBN: 978-1-4614-1583-1Published: 12 November 2011
Softcover ISBN: 978-1-4899-9976-4Published: 25 January 2014
eBook ISBN: 978-1-4614-1584-8Published: 13 November 2011
Series ISSN: 1439-7382
Series E-ISSN: 2196-9922
Edition Number: 1
Number of Pages: XIV, 278
Topics: Partial Differential Equations, Analysis, Special Functions