On the Stability of the Spectrum in the Pompeiu Problem

Let Ω be a Jordan domain in the complex plane with smooth boundary. We call the Pompeiu spectrum σ(Ω) the set of all λ such that there exists a nontrivial solution of overdetermined Dirichlet-Neumann boundary-value problem. [formula] (ν is the normal vector to the boundary ∂Ω). Let Ωt, t ∈ [0, T) be a family of Jordan domains in the complex plane with real-analytic boundaries. Suppose that Ωt analytically depends on the parameter t and Ω0 = {z ∈ C : |z| ≤ 1}. It is proved that if there exists a real-analytic function λ(t), such that λ(t) ∈ σ(Ωt), t ∈ [0, T), then all domains Ωt are discs.