Hölder gradient estimates for a class of singular or degenerate parabolic equations

Abstract

We prove interior Hölder estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equationwhere p ∈ (1, ∞) and κ ∈ (1 − p, ∞). This includes the from L ∞ to C 1,α regularity for parabolic p-Laplacian equations in both divergence form with κ = 0, and non-divergence form with κ = 2 − p. This work is a continuation of a paper by the last two authors [12].

Remarks on regularity for p-Laplacian type equations in non-divergence form

“…Indeed, one can modify the arguments used in the proof of Lemma 3.2 or use the argument in [19, proof of Theorem 4.5]. For related works, we refer to [1,19]. It would be useful to know when it is possible to provide the local regularity in time without using the regularity in space (and without assuming much regularity on the initial data) and how it would then imply the higher regularity in space.…”

“…x,t (Q 1 ), then one could adapt the argument of [19] by regularizing the equation and differentiating it, and then prove the Hölder continuity of the gradient of the solutions of (1.1) with a norm which will then depend on ||Df || L ∞ (Q 1 ) . Our study relies on a nonlinear method based on compactness arguments where we avoid differentiating the equation and assume only the continuity of f .…”

Lipschitz Regularity for a Homogeneous Doubly Nonlinear PDE

“…Recently, Ishii-Lions' method has been used for equations involving the p-Laplacian. In [IJS18], the authors used it to study the regularity of solutions of…”

Gradient continuity estimates for the normalized p-Poisson equation

“…This aspect was first studied in [33] . In recent times, the parabolic normalized p−Laplacian, as well as its degenerate and singular variants, have been studied in various contexts in a number of papers, see [1,22,13,5,6,7,18,32,19,23,21,31]. Such equations have also found applications in image processing (see for instance [13]).…”