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

14 pages