Some Results on the Schiffer's Conjecture in R^2

Let $\mathrm{\Omega }$ 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 $0 = \mu_1(\Omega) < \mu_2(\Omega) <= …$thesetofallNeumanneigenvaluesfortheLaplacianon$\Omega$,weshowthat1)if$\mu < \mu_8(\Omega)$;or2)if$\Omega$ is strictly convex and centrally symmetric, $\mu <{\mu }_{1}3(\mathrm{\Omega })$, then $\mathrm{\Omega }$ must be a disk.