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2008 Harnack estimates for quasi-linear degenerate parabolic differential equations
Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri
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Acta Math. 200(2): 181-209 (2008). DOI: 10.1007/s11511-008-0026-3


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.

Funding Statement

This work was partially supported by I.M.A.T.I.–C.N.R. (Italy).
Emmanuele DiBenedetto was supported by a NSF grant.


Dedicated to the memory of Ennio De Giorgi


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Emmanuele DiBenedetto. Ugo Gianazza. Vincenzo Vespri. "Harnack estimates for quasi-linear degenerate parabolic differential equations." Acta Math. 200 (2) 181 - 209, 2008.


Received: 20 February 2006; Published: 2008
First available in Project Euclid: 31 January 2017

zbMATH: 1221.35213
MathSciNet: MR2413134
Digital Object Identifier: 10.1007/s11511-008-0026-3

Primary: 35B65 , 35K65
Secondary: 35B45

Keywords: Degenerate parabolic equations , Harnack estimates , Hölder continuity

Rights: 2008 © Institut Mittag-Leffler


Vol.200 • No. 2 • 2008
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