| 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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