A geometric tangential approach to sharp regularity for degenerate evolution equations

Abstract

To Emmanuele DiBenedetto, with admiration and friendship, on the occasion of his 70th birthday.Abstract. We provide a broad overview on qualitative versus quantitative regularity estimates in the theory of degenerate parabolic pdes. The former relates to DiBenedetto's revolutionary method of intrinsic scaling, while the latter is achieved by means of what has been termed geometric tangential analysis. We discuss, in particular, sharp estimates for the parabolic p−Poisson equation, for the porous medium equation and for the doubly nonlinear equation.

Regularity of the p-Poisson equation in the plane

“…The topic was inspired by a talk of J. M. Urbano concerning [TU13]. G. Mingione has informed us that alternative proofs can be extracted from various estimates in [KM13].…”

A proof of the Cp′-regularity conjecture in the plane

“…This will allow us to frame the C p ′ conjecture into the formalism of the so called geometric tangential analysis, e.g. [7], [2,1] and [17,18,19,20,21,22].…”

Sharp regularity estimates for quasilinear evolution equations

“…We then iterate this estimate in a systematic manner, properly adjusted to the intrinsic scaling of equation. Inspired by the recent results from [5,6,8] and [39], we obtain an estimate (Theorem 3.2), which provides a precise control of the oscillation of weak solution of (1.1) in terms of the magnitude of its gradient.…”

“…The second compatibility condition defines the fashion in which the gradient of weak solution has a universal Hölder modulus of continuity. The regularity of weak solutions of quasilinear evolution problems received wide attention in the last decades (see, for example, [1,2,9,13,16,18,20,21,22,23,25,28,29,30,31,32,39,41,42]) due to its connection to a number of problems arising in biology, chemistry, mathematical physics, geometric and free boundary problems, etc (cf. [10] and [25] for complete essays on regularity of evolution equations with degenerate diffusion).…”

“…along the set where gradient vanishes), where the diffusivity of the equation collapses (see, for example, [15,16,17,18,37] and [38], where sharp and improved regularity estimates are obtained along the set of certain degenerate points of solutions). Furthermore, unlike [16], [19], [39], we also treat the singular case, i.e., when max 1, 2n n+2 < p < 2, which is a non-trivial task, because in this setting the degeneracy degree of p-Laplacian blows-up along critical points.…”