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

Let $\mathrm{\Omega }$ 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} -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 $\alpha$, then $\mathrm{\Omega }$ is a disk, which is the partially affirmative answer to the Berenstein conjecture. Similarly, we show that, if $\mathrm{\Omega }$ 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 $\alpha$, then $\mathrm{\Omega }$ is also a disk, which is the partially affirmative answer to the Schiffer conjecture.