Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.02724 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929676401770496 |
|---|---|
| author | Dai, Guowei |
| author_facet | Dai, Guowei |
| contents | Let $Ω$ be a bounded domain in $\mathbb{R}^{N+1}$ 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, then $Ω$ is a ball, which is exactly the 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, then $Ω$ is also a ball, which is exactly the affirmative answer to the Schiffer conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02724 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Confirmed answer to the Schiffer conjecture and the Berenstein conjecture Dai, Guowei Analysis of PDEs Let $Ω$ be a bounded domain in $\mathbb{R}^{N+1}$ 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, then $Ω$ is a ball, which is exactly the 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, then $Ω$ is also a ball, which is exactly the affirmative answer to the Schiffer conjecture. |
| title | Confirmed answer to the Schiffer conjecture and the Berenstein conjecture |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.02724 |