Confirmed answer to the Schiffer conjecture and the Berenstein conjecture

TL;DR

This paper proves that certain overdetermined elliptic boundary value problems only have solutions when the domain is a ball, confirming the Berenstein and Schiffer conjectures in these cases.

Contribution

It provides the first rigorous proof that solutions to these overdetermined problems imply the domain must be a sphere, resolving longstanding conjectures.

Findings

Domains with solutions are necessarily balls.

Confirmed the Berenstein conjecture for smooth boundaries.

Confirmed the Schiffer conjecture for Lipschitz boundaries.

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} -\Delta u=\alpha u\,\, \text{in}\,\,\Omega, \,\, u=0\,\,\text{on}\,\, \partial\Omega,\,\,\frac{\partial u}{\partial n} =c\,\,\text{on}\,\,\partial\Omega\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} -\Delta u=\alpha u\,\, \text{in}\,\,\Omega, \,\, \frac{\partial u}{\partial n}=0\,\,\text{on}\,\, \partial\Omega,\,\,u =c\,\,\text{on}\,\,\partial\Omega\nonumber \end{equation} has a nontrivial solution, then $\Omega$ is also a ball, which is exactly the affirmative answer to the Schiffer conjecture.