Acta Mathematica

Harnack estimates for quasi-linear degenerate parabolic differential equations

Emmanuele DiBenedetto, Ugo Gianazza, and Vincenzo Vespri

Full-text: Open access

Abstract

We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the p-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.

Note

Dedicated to the memory of Ennio De Giorgi

Note

This work was partially supported by I.M.A.T.I.–C.N.R. (Italy).

Emmanuele DiBenedetto was supported by a NSF grant.

Article information

Source
Acta Math. Volume 200, Number 2 (2008), 181-209.

Dates
Received: 20 February 2006
First available in Project Euclid: 31 January 2017

Permanent link to this document
http://projecteuclid.org/euclid.acta/1485891979

Digital Object Identifier
doi:10.1007/s11511-008-0026-3

Mathematical Reviews number (MathSciNet)
MR2413134

Zentralblatt MATH identifier
1221.35213

Subjects
Primary: 35K65: Degenerate parabolic equations 35B65: Smoothness and regularity of solutions
Secondary: 35B45: A priori estimates

Keywords
Degenerate parabolic equations Harnack estimates Hölder continuity

Rights
2008 © Institut Mittag-Leffler

Citation

DiBenedetto, Emmanuele; Gianazza, Ugo; Vespri, Vincenzo. Harnack estimates for quasi-linear degenerate parabolic differential equations. Acta Math. 200 (2008), no. 2, 181--209. doi:10.1007/s11511-008-0026-3. http://projecteuclid.org/euclid.acta/1485891979.


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