A geometric tangential approach to sharp regularity for degenerate evolution equations

TL;DR

This paper derives a precise formula for the Hölder continuity exponent of weak solutions to degenerate parabolic p-Laplace equations, using geometric tangential equations and intrinsic scaling methods.

Contribution

It provides the first sharp, explicit characterization of the Hölder exponent in terms of parameters p, q, r, and n for these equations.

Findings

Explicit formula for Hölder exponent α in terms of p, q, r, n.

Method based on geometric tangential equations and intrinsic scaling.

Proofs applicable to other nonlinear evolution problems.

Abstract

That the weak solutions of degenerate parabolic pdes modelled on the inhomogeneous $p-$Laplace equation $$ u_t - \mathrm{div} \left(|\nabla u|^{p-2} \nabla u \right) = f \in L^{q,r}, \quad p>2 $$ are $C^{0,\alpha}$, for some $\alpha \in (0,1)$, is known for almost 30 years. What was hitherto missing from the literature was a precise and sharp knowledge of the H\"older exponent $\alpha$ in terms of $p, q, r$ and the space dimension $n$. We show in this paper that $$ \alpha = \frac{(pq-n)r-pq}{q[(p-1)r-(p-2)]}, $$ using a method based on the notion of geometric tangential equations and the intrinsic scaling of the $p-$parabolic operator. The proofs are flexible enough to be of use in a number of other nonlinear evolution problems.