A new proof of the Hölder continuity of solutions to p-Laplace type parabolic equations

Abstract

It is a well-known fact that solutions to nonlinear parabolic partial differential equations of p-laplacian type are Hölder continuous. One of the main features of the proof, as originally given by DiBenedetto and DiBenedetto-Chen, consists in studying separately two cases, according to the size of the solution. Here we present a new proof of the Hölder continuity of solutions, which is based on the ideas used in the proof of the Harnack inequality for the same kind of equations recently given by E. DiBenedetto, U. Gianazza and V. Vespri. Our method does not rely on any sort of alternative, and has a strong geometric character.

A Quantitative Modulus of Continuity for the Two-Phase Stefan Problem

“…hence, since τ in (2.14) turns out to be, in our case, exactly τ i − τ i−1 , Harnack estimate (2.13) applied to the supersolution v i := min{v, (2c 2 ) −i k} gives 16) and the last inequality holds if inf BR 0 (x0) v i (·, τ i−1 ) ≥ (2c 2 ) −(i−1) k. Using an iterative argument, starting from (2.15), we see that (2.16) holds for any j ∈ {1, . .…”

“…The next proposition, which encodes the decay rate of supersolutions, follows from the iteration of the previous theorem; see [16,Corollary 3.4] for a very similar statement.…”

Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences

“…On the other hand, the regularity results for the full-gradient case are abundant. For the evolutionary p-case, let us restrict ourselves to referring to the classical monograph by DiBenedetto [25] and a simple proof of C 1,α loc regularity by Gianazza, Surnachev & Vespri [31].…”

Interior Harnack Estimates: The State-of-the-Art for Quasilinear Singular Parabolic Equations

“…Starting from this remark, in [34] there is an alternative proof of the Hölder regularity for degenerate quasilinear parabolic equations starting from the approach used for Harnack estimates. We recall that the corresponding alternative approach to the quasilinear singular parabolic equation can be found in [11].…”