Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Abstract

Given a eigenvalue µ 2 0m of −∆ in the unit ball B1, with Neumann boundary conditions, we prove that there exists a class D of C 0,1 -domains, depending on µ0m, such that if u is a no trivial solution to the following problem ∆u + µu = 0 in Ω, u = 0 on ∂Ω, and R ∂Ω ∂nu = 0, with Ω ∈ D, and µ = µ 2 0m + o(1), then Ω is a ball. Here µ is a eigenvalue of −∆ in Ω, with Neumann boundary conditions.