Salvato in:
| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2501.02724 |
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Sommario:
- 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.