Harnack's Inequality for Degenerate and Singular Parabolic Equations (Springer Monographs in Mathematics)

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9781461415831: Harnack's Inequality for Degenerate and Singular Parabolic Equations (Springer Monographs in Mathematics)

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.

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While degenerate and singular parabolic equations have been researched extensively for the last 25 years, the Harnack inequality for nonnegative solutions to these equations has received relatively little attention. Recent progress has been made on the Harnack inequality to the point that the theory is now reasonably complete―except for the singular subcritical range―both for the p-Laplacian and the porous medium equations.

This monograph provides a comprehensive overview of the subject that highlights open problems.  The authors treat the Harnack inequality for nonnegative solutions to p-Laplace and porous medium type equations, both in the degenerate and in the singular range. The work is mathematical in nature; its aim is to introduce a novel set of tools and techniques that deepen our understanding of the notions of degeneracy and singularity in partial differential equations.

 Although related in spirit to a monograph by the first author in this subject, this book is a self-contained treatment with a different perspective.  Here the focus is entirely on the Harnack estimates and on their applications; the authors use the Harnack inequality to reprove a number of known regularity results.  This book is aimed at researchers and advanced graduate students who work in this fascinating field.

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“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)

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