Strunk: Gradient regularity for solutions to doubly nonlinear parabolic partial differential equations

Date: 2024-05-21

Time: 16:30 - 17:00

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Michael Strunk, University of Salzburg


In this talk, we consider doubly nonlinear parabolic equations of the type

\begin{equation*}\partial_t u^q – \mathrm{div} A(x,t,Du) = 0\qquad\mbox{in } \Omega_T:=\Omega\times(0,T),\end{equation*}with \(q>0\) and \(p>1\), where the vector field \(A:\Omega_T\times\mathbb{R}^n\to\mathbb{R}^n\) satisfies the following \(p\)-growth structure conditions\begin{align*}\left\{\begin{array}{l}| A(x,t,\xi)| + (\mu^2 + |\xi|^2 )^{\frac{1}{2}}|\partial_\xi A(x,t,\xi)| \leq C_1 (\mu^2 + |\xi |^{2})^{\frac{p-1}{2}} \\[3pt]\langle \partial_{\xi}A(x,t,\xi)\eta, \eta \rangle \geq C_2 (\mu^2 + |\xi|^2 )^{\frac{p-2}{2}}|\eta|^2 \\[3pt]|\partial_x A_i (x,t,\xi)| \leq C_3 ( \mu^2 + |\xi|^2)^{\frac{p-1}{2}}\end{array}\right. \label{voraussetzungen}\end{align*}for a.e. \((x,t) \in \Omega_T\),\(i\in\{1,…,n\}\),\(\eta,\xi \in\mathbb{R}^n\),\(\mu \in [0,1]\), with positive structural constants \(C_1, C_2, C_3\). Our main result establishes the local Hölder continuity of the gradient of non-negative weak solutions in the super-critical fast diffusion regime \(0 < p-1 < q < \frac{n(p-1)}{(n-p)_+}.\) Additionally, we obtain a local bound for the spatial gradient.