  In this paper we study the Steklov-Dirichlet eigenvalues
$\lambda_k(\Omega,\Gamma_S)$, where $\Omega\subset \mathbb{R}^d$ is a domain
and $\Gamma_S\subset \partial \Omega$ 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 $\Gamma_S$, 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 $\Gamma_{S,n}$, 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.
