Sharp regularity estimates for quasilinear evolution equations

Abstract

We establish sharp geometric C 1+α regularity estimates for bounded weak solutions of evolution equations of p-Laplacian type. Our approach is based on geometric tangential methods, and makes use of a systematic oscillation mechanism combined with an adjusted intrinsic scaling argument.

Sharp regularity for degenerate obstacle type problems: A geometric approach

“…Thus, we obtain the desired estimate for all r ∈ 0, 1 4 . Finally, we obtain (8) by computing the following…”

“…Moreover, (16) assures us that u r0 is uniformly bounded in the L ∞ topology in B 1 2 . Thus, Theorem 1.10 it follows that u r0…”

“…for some σ > 0. Thus, up to a subsequence, Du k → Du uniformly in B 1 2 . In particular, from (11) we conclude that max u − φ…”

“…Araújo, Ricarte and Teixeira showed that in fact, given β ∈ (0, α F ) ∩ 0, 1 1+γ then u ∈ C 1,β B 1/2 and that this result is optimal and holds for operators that are not necessarily translation invariant. In particular, if F is convex or concave, solutions belong to C 1, 1…”

“…The aim of this work is twofold: on one hand, to get sharp regularity estimates for solutions to an obstacle type problem involving a degenerate fully nonlinear operator. On the other hand, in order to achieve such purposes, we appeal to the improvement of flatness technique and geometric estimates that have proved to be very useful in dealing with regularity issues for elliptic/parabolic PDEs in the last decade (see, [1], [3], [4], [5], [7], [12], [13], [19] and [26] for some enlightening examples), but that, to the best of the authors' scientific knowledge, have not been applied to free boundary problems of obstacle type as the one dealt with in this paper.…”