A Rigidity Result for the Schiffer Conjecture on Domain with a Hole

Let $\Omega$ be a domain with a hole containing the origin in $\mathbb{R}^2$ and $u$ be a solution to the problem where $\partial^{\pm}\Omega$ represents the outer and inner boundaries of $\Omega,$ respectively, $c$ is a constant. Let ${\mu}_k$ denote the $k{\rm th}$ Neumann eigenvalue of the Laplacian on $\Omega$ and${\Omega}_h$ is the hole. We establish that if $\mu< {\mu}_8,$ then $\Omega$ is an annulus.