Harnack's Inequality for Degenerate and Singular Parabolic Equations
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 Ha
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For further volumes: http://www.springer.com/series/3733
Emmanuele DiBenedetto Ugo Gianazza Vincenzo Vespri
Harnack’s Inequality for Degenerate and Singular Parabolic Equations
123
Emmanuele DiBenedetto Department of Mathematics Vanderbilt University Nashville, 37240 TN USA [email protected]
Ugo Gianazza Dipartimento di Matematica “F. Casorati” Università degli Studi di Pavia 27100 Pavia Italy [email protected]
Vincenzo Vespri Dipartimento di Matematica “U. Dini” Università degli Studi di Firenze 50134 Firenze Italy [email protected]
ISSN 1439-7382 ISBN 978-1-4614-1583-1 e-ISBN 978-1-4614-1584-8 DOI 10.1007/978-1-4614-1584-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941123
Mathematics Subject Classification (2010): 35-XX, 35K65, 35K67, 35J62
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 The Classical Harnack Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Quasilinear Coercive Elliptic and Parabolic Equations . . . . . . . . 4 3 Degenerate and Singular Parabolic Equations . . . . . . . . . . . . . . . 5 3.1 Quasilinear Equations of p-Laplacian Type . . . . . . . . . . . 5 3.2 Quasilinear Equations of Porous Medium Type . . . . . . . . 6 3.3 Aim of the Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Parabolic Harnack Estimates. The Role of the Structure . . . . . . 7 4.1 Degenerate Equations of the p-Laplacian Type for p > 2 8 4.2 Singular Equations of the p-Laplacian Type for 2N 9 N +1 < p < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Outstanding Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Poincar´e and Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 2 Cuts and Truncations of Functions in W 1,p (E) and Their Embeddings . . . . . . . . . . . . . . . . .
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