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 $$ u_t - \mathrm{div} \left(|\nabla u|^{p-2} \nabla u \right) = f \in L^{q,r}, \quad p>2 $$ 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.