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

Abstract

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, $\mu <{\mu }_{1}3(\Omega )$, then $\Omega$ must be a disk.

12 pages, third version, accepted by JDE