A Rigidity Result for the Schiffer Conjecture on Domain with a Hole (Analysis in Theory and Applications, 2025)
Let $Ω$ be a domain with a hole containing the origin in $\mathbb{R}^2$ and $u$ be a solution to the problem where $\partial^{±}Ω$ represents the outer and inner boundaries of $Ω,$ respectively, $c$ is a constant. Let ${μ}_k$ denote the $k{\rm th}$ Neumann eigenvalue of the Laplacian on $Ω$ and${Ω}_h$ is the hole. We establish that if $μ< μ_8,$ then $Ω$ is an annulus.
Keywords: Schiffer conjecture; overdetermined problem; symmetry.