Gradient regularity for widely degenerate parabolic equations

Abstract

In this paper, we are interested in the regularity of weak solutions $u\colon\Omega_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $\Omega_T$}, \end{equation*} where $\mathcal{F}$ is only elliptic for values of $Du$ outside a bounded and convex set $E\subset \mathbb{R}^n$ with the property that $0\in \mathrm{Int}{E}$. Here, $\Omega_T :=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$ denotes a space-time cylinder taken over a bounded domain $\Omega\subset\mathbb{R}^n$ for some finite time $T>0$. The function $\mathcal{F} : \Omega_T\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ present in the diffusion is assumed to satisfy: the partial mapping $\xi\mapsto \mathcal{F}(x,t,\xi)$ is regular whenever $\xi$ lies outside of $E$, and vanishes entirely whenever $\xi$ lies within this set. Additionally, the datum $f$ is assumed to be of class $L^{n+2+\sigma}(\Omega_T)$ for some parameter $\sigma > 0$. As our main result we establish that \begin{equation*} \mathcal{K}(Du)\in C^0(\Omega_T) \end{equation*} for any continuous function $\mathcal{K}\in C^0(\mathbb{R}^n)$ that vanishes on $E$. This article aims to extend the $C^1$-regularity result for the elliptic case to the parabolic setting.