Unified approach to $C^{1,\alpha}$ regularity for quasilinear parabolic equations

TL;DR

This paper presents a unified method to establish $C^{1,eta}$ regularity for weak solutions of quasilinear parabolic equations, avoiding separate treatment of singular and degenerate cases through innovative scaling and covering techniques.

Contribution

The authors develop a novel unified approach for $C^{1,eta}$ regularity estimates that applies to all cases of quasilinear parabolic equations without case-by-case analysis.

Findings

Unified $C^{1,eta}$ regularity estimates achieved

Method applies to singular and degenerate cases simultaneously

Enhanced understanding of regularity for quasilinear parabolic equations

Abstract

In this paper, we are interested in obtaining a unified approach for $C^{1,\alpha}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being \[ u_t - \text{div} (|\nabla u|^{p-2} \nabla u) = 0. \] without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a delicate adaptation of the covering argument developed by E.~DiBenedetto and A.~Friedman.