| Abstract: |
| In this talk, we consider parabolic systems of $p$-Laplace type
\begin{equation*}
\partial_tu-\Div\Big( a(x,t)\big(\mu^2+|Du|^2\big)^\frac{p-2}2Du\Big)=0
\qquad\mbox{in $E_T$},
\end{equation*}
where $p>1$, $\mu\in[0,1]$, and the coefficient $a\in L^\infty(E_T)$ is bounded below by a positive constant and H\older continuous with respect to the spatial variable $x$. Via Schauder estimates we establish local H\older continuity of the spatial gradient of bounded weak solutions.
As an application, we derive H\older estimates for the gradient of weak solutions to a doubly nonlinear parabolic equation in the supercritical fast diffusion regime. In particular, we obtain quantitative bounds for the spatial gradient and its H\older continuity. This is joint work with F. Duzaar, U. Gianazza, N. Liao, and C. Scheven. |
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