Steklov-Dirichlet spectrum: stability, optimization and continuity of eigenvalues

Abstract

In this paper we study the Steklov-Dirichlet eigenvalues λ k (Ω, ΓS), where Ω ⊂ R d is a domain and ΓS ⊂ ∂Ω is the subset of the boundary in which we impose the Steklov conditions. After a first discussion about the regularity properties of the Steklov-Dirichlet eigenfunctions we obtain a stability result for the eigenvalues. We study the optimization problem under a measure constraint on the set ΓS, we prove the existence of a minimizer and the non-existence of a maximizer. In the plane we prove a continuity result for the eigenvalues imposing a bound on the number of connected components of the sequence ΓS,n, obtaining in this way a version of the famous result of V. Šverák ([22]) for the Steklov-Dirichlet eigenvalues. Using this result we prove the existence of a maximizer under the same topological constraint and the measure constraint.