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Harnack's Inequality for Degenerate and Singular Parabolic Equations

  • Emmanuele DiBenedetto
  • Ugo Gianazza
  • Vincenzo Vespri

Part of the Springer Monographs in Mathematics book series (SMM)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 1-11
  3. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 13-22
  4. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 23-55
  5. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 57-90
  6. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 91-136
  7. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 137-185
  8. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 187-204
  9. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 205-219
  10. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 221-253
  11. Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
    Pages 255-267
  12. Back Matter
    Pages 269-278

About this book

Introduction

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.

Keywords

Harnack's inequality classical Harnack inequality degenerate and singular parabolic equations degenerate parabolic equations homogeneous monotone singular equations quasilinear parabolic equations singular parabolic equations

Authors and affiliations

  • Emmanuele DiBenedetto
    • 1
  • Ugo Gianazza
    • 2
  • Vincenzo Vespri
    • 3
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA
  2. 2., Dipt. di Matematica "F. Casorati"Universita' degli Studi di PaviaPaviaItaly
  3. 3., Dipartimento di MatematicaUniversità degli studi di FirenzeFirenzeItaly

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4614-1584-8
  • Copyright Information Springer Science+Business Media, LLC 2012
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4614-1583-1
  • Online ISBN 978-1-4614-1584-8
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site