, Volume 200, Issue 2, pp 181-209

Harnack estimates for quasi-linear degenerate parabolic differential equations

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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.

Dedicated to the memory of Ennio De Giorgi
This work was partially supported by I.M.A.T.I.–C.N.R. (Italy).
Emmanuele DiBenedetto was supported by a NSF grant.