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

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

are $C^{0,\alpha }$, for some $\alpha \in \left(0,1\right)$, has been 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

$$\alpha =\frac{\left(pq-n\right)r-pq}{q\left[\left(p-1\right)r-\left(p-2\right)\right]}$$

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.