Some results on the Schiffer conjecture in R2R2

Let Ω   be an open, bounded domain in R2R2 with connected and C∞C∞ boundary, and ω a solution ofequation(0.1)−△ω=μω,−△ω=μω,equation(0.2)∂ω∂n|∂Ω=0,equation(0.3)ω|∂Ω=const≠0ω|∂Ω=const≠0 for some μ>0μ>0. Denoting by 0=μ1(Ω)<μ2(Ω)⩽⋯0=μ1(Ω)<μ2(Ω)⩽⋯ the set of all Neumann eigenvalues for the Laplacian on Ω  , we show that 1) if μ<μ8(Ω)μ<μ8(Ω); or 2) if Ω   is strictly convex and centrally symmetric, μ<μ13(Ω)μ<μ13(Ω), then Ω must be a disk.