# A Boundary Estimate for Quasi-Linear Diffusion Equations

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Let $E$ be an open set in $\mathbb{R}^N$, and for $T>0$ let $E_T$ denote the cylindrical domain $E\times[0,T]$. We consider quasi-linear, parabolic partial differential equations of the form $$ u_t-\operatorname{div}\textbf{A}(x,t,u, Du) = 0\quad \text{ weakly in }\> E_T, $$ where the function $\textbf{A}(x,t,u,\xi)\colon E_T\times\mathbb{R}^{N+1}\to\mathbb{R}^N$ is assumed to be measurable with respect to $(x, t) \in E_T$ for all $(u,\xi)\in\mathbb{R}\times\mathbb{R}^N$, and continuous with respect to $(u,\xi)$ for a.e.~$(x,t)\in E_T$. Moreover, we assume the structure conditions $$\begin{cases} \textbf{A}(x,t,u,\xi)\cdot \xi\geq C_0|\xi|^p,&\\ \textbf{A}(x,t,u,\xi)|\leq C_1|\xi|^{p-1},& \end{cases}$$ for a.e. $(x,t)\in E_T$, $\forall\,u\in\mathbb{R},\,\forall\xi\in\mathbb{R}^N$, where $C_0$ and $C_1$ are given positive constants, and we take $p>1$. We consider a boundary datum $g$ with $$\begin{cases} g\in L^p\big(0,T;W^{1,p}( E)\big),&\\ g \text{ continuous on}\ \overline{E}_T\ \text{with modulus of continuity }\ \omega_g(\cdot),& \end{cases}$$ and we are interested in the boundary behavior of solutions to the Cauchy-Dirichlet problem $$\begin{cases} u_t-\operatorname{div}\textbf{A}(x,t,u, Du) = 0&\text{ weakly in }\> E_T,\\ u(\cdot,t)\Big|_{\partial E}=g(\cdot,t)&\text{ a.e. }\ t\in(0,T],\\ u(\cdot,0)=g(x,0),& \end{cases}$$ with $g$ as above. We do not impose any {\em a priori} requirements on the boundary of the domain $E\subset\mathbb{R}^N$, and we provide an estimate on the modulus of continuity at a boundary point in terms of a Wiener-type integral, defined by a proper elliptic $p$-capacity. The results depend on the value of $p$, namely whether $1<p\le\frac{2N}{N+1}$, $\frac{2N}{N+1}<p<2$, $p\ge2$.

This is a joint work with Naian Liao (Salzburg University, Austria) and Teemu Lukkari (Aalto University, Finland).