A geometric tangential approach to sharp regularity for degenerate evolution equations

That the weak solutions of degenerate parabolic PDEs modelled on the inhomogeneous p-Laplace equation ut-div(|∇u|p-2∇u)= f ∈ Lq,r, p≥2 are C0,α for some α ∈ (0,1) has been known for almost 30 years. What was hitherto missing from the literature was a precise and sharp knowledge of the Hölder exponent α in terms of p, q, r and the space dimension n. We show in this paper that 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. © 2014 Mathematical Sciences Publishers.