A geometric tangential approach to sharp regularity for degenerate evolution equations

Abstract

That the weak solutions of degenerate parabolic pdes modelled on the inhomogeneous p−Laplace equation ut − div ( |∇u|p−2∇u ) = f ∈ L, p > 2 are C0,α, for some α ∈ (0, 1), is known for almost 30 years. What was hitherto missing from the literature was a precise and sharp knowledge of the Holder exponent α in terms of p, q, r and the space dimension n. We show in this paper that α = (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.