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

TL;DR

Establishes that in the plane, any bounded convex centrally symmetric domain with either $C^{2,\varepsilon}$ or Lipschitz boundary that admits a nontrivial solution to the overdetermined eigenvalue problem for sufficiently large $\alpha$ must be a disk. The approach hinges on a finite oscillatory-integral expansion for plane-wave probes and a new Leibniz-type formula for the operator $\Box$, enabling a single large $\alpha$ to yield constant breadth. From constant breadth and central symmetry, the boundary is shown to have constant distance from the center, hence $\partial\Omega$ is a circle. This yields partial affirmations of both Schiffer and Berenstein conjectures in 2D and introduces a technique with potential applicability to Pompeiu-type questions.

Abstract

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

Theorems & Definitions (2)

• Lemma 3.1
• proof