#### Vol. 7, No. 3, 2014

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A geometric tangential approach to sharp regularity for degenerate evolution equations

### Eduardo V. Teixeira and José Miguel Urbano

Vol. 7 (2014), No. 3, 733–744
##### Abstract

That the weak solutions of degenerate parabolic PDEs modelled on the inhomogeneous $p$-Laplace equation

${u}_{t}-÷\left(|\nabla u{|}^{p-2}\nabla u\right)=f\in {L}^{q,r},\phantom{\rule{1em}{0ex}}p>2$

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.

##### Keywords
degenerate parabolic equations, sharp Hölder regularity, tangential equations, intrinsic scaling
##### Mathematical Subject Classification 2010
Primary: 35K55, 35K65, 35B65