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

Let $Ω$ be an open, bounded domain in the plane with connected and smooth boundary, and $ω$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $μ> 0$. If the boundary value of $ω$ is a nonzero constant along the boundary, denoting $0 = μ_1(Ω) < μ_2(Ω) <= ...$ the set of all Neumann eigenvalues for the Laplacian on $Ω$, we show that 1) if $μ< μ_8(Ω)$; or 2) if $Ω$ is strictly convex and centrally symmetric, $μ< μ_13(Ω)$, then $Ω$ must be a disk.

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