Gradient regularity for widely degenerate elliptic partial differential equations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913892010033152 |
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| author | Strunk, Michael |
| author_facet | Strunk, Michael |
| contents | In this paper, we investigate the regularity of weak solutions $u\colonΩ\to\mathbb{R}$ to elliptic equations of the type \begin{equation*}
\mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $Ω$}, \end{equation*} whose ellipticity degenerates in a fixed bounded and convex set $E\subset\mathbb{R}^n$ with $0\in \mathrm{Int}\, E$. Here, $Ω\subset\mathbb{R}^n$ denotes a bounded domain, and $\mathcal{F} \colon Ω\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ is a function with the properties: for any $x\inΩ$, the mapping $ξ\mapsto \mathcal{F}(x,ξ)$ is regular outside $E$ and vanishes entirely within this set. Additionally, we assume $f\in L^{n+σ}(Ω)$ for some $σ> 0$, representing an arbitrary datum. Our main result establishes the regularity
\begin{equation*}
\mathcal{K}(Du)\in C^0(Ω)
\end{equation*} for any continuous function $\mathcal{K}\in C^0(\mathbb{R}^n)$ vanishing on $E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient regularity for widely degenerate elliptic partial differential equations Strunk, Michael Analysis of PDEs In this paper, we investigate the regularity of weak solutions $u\colonΩ\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $Ω$}, \end{equation*} whose ellipticity degenerates in a fixed bounded and convex set $E\subset\mathbb{R}^n$ with $0\in \mathrm{Int}\, E$. Here, $Ω\subset\mathbb{R}^n$ denotes a bounded domain, and $\mathcal{F} \colon Ω\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ is a function with the properties: for any $x\inΩ$, the mapping $ξ\mapsto \mathcal{F}(x,ξ)$ is regular outside $E$ and vanishes entirely within this set. Additionally, we assume $f\in L^{n+σ}(Ω)$ for some $σ> 0$, representing an arbitrary datum. Our main result establishes the regularity \begin{equation*} \mathcal{K}(Du)\in C^0(Ω) \end{equation*} for any continuous function $\mathcal{K}\in C^0(\mathbb{R}^n)$ vanishing on $E$. |
| title | Gradient regularity for widely degenerate elliptic partial differential equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2506.11708 |