Some results on the Schiffer conjecture in R2

AbstractLet Ω be an open, bounded domain in R2 with connected and C∞ boundary, and ω a solution of(0.1)−△ω=μω,(0.2)∂ω∂n|∂Ω=0,(0.3)ω|∂Ω=const≠0 for some μ>0. Denoting by 0=μ1(Ω)<μ2(Ω)⩽⋯ the set of all Neumann eigenvalues for the Laplacian on Ω, we show that 1) if μ<μ8(Ω); or 2) if Ω is strictly convex and centrally symmetric, μ