Consider the following overdetermined eigenvalue problem for $\Omega \subset \Bbb{R}^2$: $$(1) \ \ \ \ \begin{cases} - \Delta u = \lambda u & \text{ in }\Omega \\ u = 0 & \text{ on }\partial \Omega \\ \displaystyle\frac{\partial u}{\partial n} = c & \text{ on }\partial\Omega \end{cases} $$ where $c\neq 0$ is a constant. It is known that if we add the condition $u>0$ (which corresponds to the case of the first eigenfunction) then this problem can have a solution only if $\Omega$ is a disk (I'm not sure if this is true in higher dimensions, but I think it is).
The proof of the above fact relies on $u>0$, so it cannot be adapted for higher eigenvalues. Stil, there is a result of Bernstein (link), which says that if the following overdetermined problem has a solution for an infinite number of $\lambda$, then $\Omega$ is a disk.
The following conjecture arises:
Conjecture: If $(1)$ has a solution, then $\Omega$ is a disk.
My questions are:
- Has this conjecture been (partially) proved?
- Has any counter-example been found?
For 1. I found an article of Dalmasso Robert which solves $(1)$ for $\lambda \leq \lambda_2$ and $\Omega$ convex and symmetric. I found other reference, by Jian Deng, where the Neumann variant (Schiffer conjecture) is studied.
