Confirmed answer to the Schiffer conjecture and the Berenstein conjecture

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{N+1}$ with a connected $C^{2,\epsilon }$ ($\epsilon \in (0,1)$) 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 $\Omega$ is a ball, which is exactly the affirmative answer to the Berenstein conjecture. Similarly, we show that, if $\Omega$ 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 $\Omega$ is also a ball, which is exactly the affirmative answer to the Schiffer conjecture.

arXiv commentThe proof of constant breadth of domain is insufficient. After the modification, I will upload it again