Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations. (English) Zbl 1178.35100

Kuusi, Tuomo

Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 7, No. 4, 673-716 (2008).

The author proves both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. A similar result was proved by DiBenedetto, Gianazza and Vespri by using extensively De Giorgi’s estimates. Here the author follows the original approach due to Moser and Trudinger and he reduces the proof to an analysis of so-called hot and cold alternatives, and uses the expansion of positivity togheter with a parabolic type of Krylov-Safonov covering argument. We stress that the author uses only the properties of weak supersolutions and does not use any comparison to weak solutions.

Reviewer: Vincenzo Vespri (Firenze)

MSC:

35B45	A priori estimates in context of PDEs
35K65	Degenerate parabolic equations
35K20	Initial-boundary value problems for second-order parabolic equations
35K59	Quasilinear parabolic equations

Keywords:

degenerate \(p\)-Laplacean; Moser’s and Trudinger approach; hot and cold alternatives; Krylov-Safonov covering argument