Some Results on the Schiffer's Conjecture in R²

Let $\Omega$ be an open, bounded domain in the plane with connected and smooth boundary, and $\omega$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $\mu > 0$. If the boundary value of $\omega$ is a nonzero constant along the boundary, denoting $0 = \mu_1(\Omega) < \mu_2(\Omega) <= ...$ the set of all Neumann eigenvalues for the Laplacian on $\Omega$, we show that 1) if $\mu < \mu_8(\Omega)$; or 2) if $\Omega$ is strictly convex and centrally symmetric, $\mu < \mu_13(\Omega)$, then $\Omega$ must be a disk.