On the Stability of the Spectrum in the Pompeiu Problem

Abstract

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 Ω t , t ∈ [0, T) be a family of Jordan domains in the complex plane with real-analytic boundaries. Suppose that Ω t analytically depends on the parameter t and Ω 0 = { z ∈ C : | z| ≤ 1}. It is proved that if there exists a real-analytic function λ( t), such that λ( t) ∈ σ( Ω t ), t ∈ [0, T), then all domains Ω t are discs.