On the Schiffer and Berenstein conjectures for centrally symmetric convex domains in the plane
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915636578353152 |
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| author | Dai, Guowei Sun, Yingxin Wei, Juncheng Zhang, Yong |
| author_facet | Dai, Guowei Sun, Yingxin Wei, Juncheng Zhang, Yong |
| contents | Let $Ω$ be a bounded, convex, centrally symmetric in $\mathbb{R}^{2}$ with a connected $C^{2,ε}$ ($ε\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -Δu=αu\,\, \text{in}\,\,Ω, \,\, u=0\,\,\text{on}\,\, \partialΩ,\,\,\frac{\partial u}{\partial n} =c\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} has a nontrivial solution corresponding to a sufficiently large eigenvalue $α$, then $Ω$ is a disk, which is the partially affirmative answer to the Berenstein conjecture. Similarly, we show that, if $Ω$ has a Lipschitz connected boundary and the following overdetermined elliptic problem \begin{equation} -Δu=αu\,\, \text{in}\,\,Ω, \,\, \frac{\partial u}{\partial n}=0\,\,\text{on}\,\, \partialΩ,\,\,u =c\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} has a nontrivial solution corresponding to a sufficiently large eigenvalue $α$, then $Ω$ is also a disk, which is the partially affirmative answer to the Schiffer conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19819 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Schiffer and Berenstein conjectures for centrally symmetric convex domains in the plane Dai, Guowei Sun, Yingxin Wei, Juncheng Zhang, Yong Analysis of PDEs Let $Ω$ be a bounded, convex, centrally symmetric in $\mathbb{R}^{2}$ with a connected $C^{2,ε}$ ($ε\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -Δu=αu\,\, \text{in}\,\,Ω, \,\, u=0\,\,\text{on}\,\, \partialΩ,\,\,\frac{\partial u}{\partial n} =c\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} has a nontrivial solution corresponding to a sufficiently large eigenvalue $α$, then $Ω$ is a disk, which is the partially affirmative answer to the Berenstein conjecture. Similarly, we show that, if $Ω$ has a Lipschitz connected boundary and the following overdetermined elliptic problem \begin{equation} -Δu=αu\,\, \text{in}\,\,Ω, \,\, \frac{\partial u}{\partial n}=0\,\,\text{on}\,\, \partialΩ,\,\,u =c\,\,\text{on}\,\,\partialΩ\nonumber \end{equation} has a nontrivial solution corresponding to a sufficiently large eigenvalue $α$, then $Ω$ is also a disk, which is the partially affirmative answer to the Schiffer conjecture. |
| title | On the Schiffer and Berenstein conjectures for centrally symmetric convex domains in the plane |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.19819 |