Global gradient bounds for the parabolicp-Laplacian system

Abstract

ABSTRACT. A by now classical result due to DiBenedetto states that the spatial gradient of solutions to the parabolic p-Laplacian system is locally Hölder continuous in the interior. However, the boundary regularity is not yet well understood. In this paper we prove a boundary L ∞ -estimate for the spatial gradient Du of solutions to the parabolic p-Laplacian systemfor p ≥ 2, together with a quantitative estimate. In particular, this implies the global Lipschitz regularity of solutions. The result continues to hold for the so called asymptotically regular parabolic systems.

Continuity of solutions to a nonlinear fractional diffusion equation

“…and observing that R i − 4 h 0 = R i+1 + 4 h 0 , we obtain the iterative scheme of inequalities: 2 We observe that by construction we have…”

“…We refer to [12] and [13] for a complete account on the regularity results for this equation and some of its generalizations. At present, the best local regularity known is spatial C 1,α −regularity for some α > 0 (see [12,Chapter IX]) and C 0,1/2 −regularity in time (see [2,Theorem 2.3]). None of these exponents is known to be sharp.…”

Parabolic p-Laplacian revisited: Global regularity and fractional smoothness

“…Concerning results that are somehow connected to Theorem 1, without the intention of being complete, we may cite [1,5,15,20,21,[26][27][28] and the references therein. Let us comment and connect these results to our theorem.…”

“…In what regards higher integrability to the gradients, these results were recently generalized to global space bounds for parabolic systems which admit the p-Laplacian case, when f ≡ 0 and nonhomogeneous boundary data is considered, cf. [5]. For homogeneous boundary data in very rough domains, in [9] the authors provide global Calderón-Zygmund estimates for a class of degenerate parabolic equations.…”

“…There have been several contributions related to mixed space-time regularity for solutions of a broad class of parabolic degenerate equations or systems. For instance, the reader is referred to [1,5,9,13,14,23,25], and the references therein, where beautiful contributions for the theory are provided. However, there are still few results on the gain of time or space regularity of the solutions with respect to vector-valued Banach spaces.…”