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

In this paper we study the Steklov-Dirichlet eigenvalues $λ_k(Ω,Γ_S)$, where $Ω\subset \mathbb{R}^d$ is a domain and $Γ_S\subset \partial Ω$ 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 $Γ_S$, 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 $Γ_{S,n}$, obtaining in this way a version of the famous result of V. Sverak for the Steklov-Dirichlet eigenvalues. Using this result we prove the existence of a maximizer under the same topological constraint and the measure constraint.