Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations
Authors:
Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri
Journal:
Electron. Res. Announc. Amer. Math. Soc. 12 (2006), 95-99
MSC (2000):
Primary 35K65, 35B65; Secondary 35B45
DOI:
https://doi.org/10.1090/S1079-6762-06-00166-1
Published electronically:
August 2, 2006
MathSciNet review:
2237273
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Abstract: We establish the intrinsic Harnack inequality for nonnegative solutions of the parabolic $p$-Laplacian equation by a proof that uses neither the comparison principle nor explicit self-similar solutions. The significance is that the proof applies to quasilinear $p$-Laplacian-type equations, thereby solving a long-standing problem in the theory of degenerate parabolic equations.
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- Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384
dgv E. DiBenedetto, U. Gianazza, and V. Vespri, Local clustering on the nonzero set of functions in $W^{1,1}_{loc}(E)$, Rend. Lincei Mat. Appl. 17 (2006), 223–225.
dgv2 E. DiBenedetto, U. Gianazza, and V. Vespri, Harnack estimates for quasilinear degenerate parabolic differential equations, in preparation.
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ar-serr D.G. Aronson and J. Serrin, Local behaviour of solutions of quasi-linear parabolic equations, Arch. Rat. Mech. Anal. 25 (1967), 81–123.
dibe-mod E. DiBenedetto, Harnack estimates in certain function classes, Atti Sem. Mat. Fis. Univ. Modena 37 (1989), 173–182.
dibe-sv E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Series Universitext, New York, 1993.
dgv E. DiBenedetto, U. Gianazza, and V. Vespri, Local clustering on the nonzero set of functions in $W^{1,1}_{loc}(E)$, Rend. Lincei Mat. Appl. 17 (2006), 223–225.
dgv2 E. DiBenedetto, U. Gianazza, and V. Vespri, Harnack estimates for quasilinear degenerate parabolic differential equations, in preparation.
hadamard J. Hadamard, Extension à l’équation de la chaleur d’un théorème de A. Harnack, Rend. Circ. Mat. di Palermo, Ser. 2, 3 (1954), 337–346.
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trud N.S. Trudinger, Pointwise estimates and quasi-linear parabolic equations, Comm. Pure Appl. Math. 21 (1968), 205–226.
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Additional Information
Emmanuele DiBenedetto
Affiliation:
Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN 37240, USA
Email:
em.diben@vanderbilt.edu
Ugo Gianazza
Affiliation:
Dipartimento di Matematica “F. Casorati", Università di Pavia, via Ferrata 1, 27100 Pavia, Italy
ORCID:
0000-0003-2558-560X
Email:
gianazza@imati.cnr.it
Vincenzo Vespri
Affiliation:
Dipartimento di Matematica “U. Dini", Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
Email:
vespri@math.unifi.it
Keywords:
Degenerate parabolic equations,
Harnack estimates,
Hölder continuity
Received by editor(s):
January 20, 2006
Published electronically:
August 2, 2006
Communicated by:
Luis A. Caffarelli
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
