Modica type estimates and curvature results for overdetermined $p$-Laplace problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913898869817344 |
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| author | Lian, Yuanyuan Wu, Jing |
| author_facet | Lian, Yuanyuan Wu, Jing |
| contents | In this paper we prove Modica type estimates for the following overdetermined $p$-Laplace problem \begin{equation*}
\begin{cases}
\mathrm{div} \left(|\nabla u|^{p-2}\nabla u\right)+f(u) =0& \mbox{in $Ω$, }
u>0 &\mbox{in $Ω$, }
u=0 &\mbox{on $\partialΩ$, }
\partial_ν u=-κ&\mbox{on $\partialΩ$, }
\end{cases} \end{equation*} where $1<p<+\infty$, $f\in C^1(\mathbb{R})$, $Ω\subset \mathbb{R}^n$ ($n\geq 2$) is a $C^1$ domain (bounded or unbounded), $ν$ is the exterior unit normal of $\partial Ω$ and $κ\geq 0$ is a constant. Based on Modica type estimates, we obtain rigidity results for bounded solutions. In particular, we prove that if there exists a nonpositive primitive $F$ of $f$ satisfying $F(0)\geq -(p-1)κ^p / p$ (for $p>2$ we also assume that if $F(u_0)=0$, $F(u)=O(|u-u_0|^p)$ as $u\rightarrow u_0$), then either the mean curvature of $\partial Ω$ is strictly negative or $Ω$ is a half-space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14579 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modica type estimates and curvature results for overdetermined $p$-Laplace problems Lian, Yuanyuan Wu, Jing Analysis of PDEs 35N25, 35B50, 35J92 In this paper we prove Modica type estimates for the following overdetermined $p$-Laplace problem \begin{equation*} \begin{cases} \mathrm{div} \left(|\nabla u|^{p-2}\nabla u\right)+f(u) =0& \mbox{in $Ω$, } u>0 &\mbox{in $Ω$, } u=0 &\mbox{on $\partialΩ$, } \partial_ν u=-κ&\mbox{on $\partialΩ$, } \end{cases} \end{equation*} where $1<p<+\infty$, $f\in C^1(\mathbb{R})$, $Ω\subset \mathbb{R}^n$ ($n\geq 2$) is a $C^1$ domain (bounded or unbounded), $ν$ is the exterior unit normal of $\partial Ω$ and $κ\geq 0$ is a constant. Based on Modica type estimates, we obtain rigidity results for bounded solutions. In particular, we prove that if there exists a nonpositive primitive $F$ of $f$ satisfying $F(0)\geq -(p-1)κ^p / p$ (for $p>2$ we also assume that if $F(u_0)=0$, $F(u)=O(|u-u_0|^p)$ as $u\rightarrow u_0$), then either the mean curvature of $\partial Ω$ is strictly negative or $Ω$ is a half-space. |
| title | Modica type estimates and curvature results for overdetermined $p$-Laplace problems |
| topic | Analysis of PDEs 35N25, 35B50, 35J92 |
| url | https://arxiv.org/abs/2506.14579 |