On the Schiffer and Berenstein conjectures for centrally symmetric convex domains in the plane

TL;DR

This paper proves that certain overdetermined elliptic problems in convex, centrally symmetric planar domains imply the domain must be a disk, providing partial answers to the Schiffer and Berenstein conjectures.

Contribution

It establishes that large eigenvalue solutions to specific overdetermined elliptic problems force the domain to be a disk, advancing the understanding of these conjectures.

Findings

Solutions exist only for disk-shaped domains under given conditions.

Confirms the Berenstein conjecture for smooth boundary domains.

Supports the Schiffer conjecture for Lipschitz boundary domains.

Abstract

Let $\Omega$ be a bounded, convex, centrally symmetric in $\mathbb{R}^{2}$ 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 corresponding to a sufficiently large eigenvalue $\alpha$, then $\Omega$ is a disk, which is the partially 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 corresponding to a sufficiently large eigenvalue $\alpha$, then $\Omega$ is also a disk, which is the partially affirmative answer to the Schiffer conjecture.