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In this paper we study the Steklov-Dirichlet eigenvalues$\lambda_k(\Omega,\Gamma_S)$, where $\Omega\subset \mathbb{R}^d$ is a domainand $\Gamma_S\subset \partial \Omega$ is the subset of the boundary in which weimpose the Steklov conditions. After a first discussion about the regularityproperties of the Steklov-Dirichlet eigenfunctions we obtain a stability resultfor the eigenvalues. We study the optimization problem under a measureconstraint on the set $\Gamma_S$, we prove the existence of a minimizer and thenon-existence of a maximizer. In the plane we prove a continuity result for theeigenvalues imposing a bound on the number of connected components of thesequence $\Gamma_{S,n}$, obtaining in this way a version of the famous resultof V. Sverak for the Steklov-Dirichlet eigenvalues. Using this result we provethe existence of a maximizer under the same topological constraint and themeasure constraint.

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