Harnack estimates for quasi-linear degenerate parabolic differential equations

2008/01/01 by Emmanuele DiBenedetto, Ugo Gianazza, Vincenzo Vespri · 6 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · pdf · doi:10.1007/s11511-008-0026-3

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.

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• On the Hölder regularity of signed solutions to a doubly nonlinear equation
• Harnack's inequality for doubly nonlinear equations of slow diffusion type. [europepmc]

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