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

Abstract

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