Gradient regularity for widely degenerate parabolic equations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916998483542016 |
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| author | Strunk, Michael |
| author_facet | Strunk, Michael |
| contents | In this paper, we are interested in the regularity of weak solutions $u\colonΩ_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*}
\partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $Ω_T$}, \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, $Ω_T :=Ω\times(0,T)\subset\mathbb{R}^{n+1}$ denotes a space-time cylinder taken over a bounded domain $Ω\subset\mathbb{R}^n$ for some finite time $T>0$. The function $\mathcal{F} : Ω_T\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ present in the diffusion is assumed to satisfy: the partial mapping $ξ\mapsto \mathcal{F}(x,t,ξ)$ is regular whenever $ξ$ lies outside of $E$, and vanishes entirely whenever $ξ$ lies within this set. Additionally, the datum $f$ is assumed to be of class $L^{n+2+σ}(Ω_T)$ for some parameter $σ> 0$. As our main result we establish that
\begin{equation*}
\mathcal{K}(Du)\in C^0(Ω_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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient regularity for widely degenerate parabolic equations Strunk, Michael Analysis of PDEs In this paper, we are interested in the regularity of weak solutions $u\colonΩ_T\to\mathbb{R}$ to parabolic equations of the type \begin{equation*} \partial_t u - \mathrm{div} \nabla \mathcal{F}(x,t,Du) = f\qquad\mbox{in $Ω_T$}, \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, $Ω_T :=Ω\times(0,T)\subset\mathbb{R}^{n+1}$ denotes a space-time cylinder taken over a bounded domain $Ω\subset\mathbb{R}^n$ for some finite time $T>0$. The function $\mathcal{F} : Ω_T\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ present in the diffusion is assumed to satisfy: the partial mapping $ξ\mapsto \mathcal{F}(x,t,ξ)$ is regular whenever $ξ$ lies outside of $E$, and vanishes entirely whenever $ξ$ lies within this set. Additionally, the datum $f$ is assumed to be of class $L^{n+2+σ}(Ω_T)$ for some parameter $σ> 0$. As our main result we establish that \begin{equation*} \mathcal{K}(Du)\in C^0(Ω_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. |
| title | Gradient regularity for widely degenerate parabolic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.07999 |