Estimates for the first and second Steklov-Dirichlet eigenvalues

In this paper, we deal with the Steklov-Dirichlet eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $\Omega_r = \Omega_0 \setminus \overline{B}_r$, where $\Omega_0 \subset \mathbb{R}^n$, $n \geq 2$, is an open, bounded set with a Lipschitz boundary, and $B_r$ is the ball centered at the origin with radius $r > 0$, such that $\overline{B}_r \subset \Omega_0$. In the first part of the paper, we focus on the first Steklov-Dirichlet eigenvalue $\sigma_1(\Omega_r)$ and prove that the sequence of corresponding normalized eigenfunctions converges to a particular constant as $r \to 0^+$. This will allow us to prove an isoperimetric inequality for $\sigma_1(\Omega_r)$ when $r$ is small enough, under a measure constraint. The second part is focused on the second Steklov-Dirichlet eigenvalue $\sigma_2(\Omega_r)$. We prove that it converges to the first non-trivial Steklov eigenvalue $\overline{\sigma}_1(\Omega_0)$ of the non-perforated domain $\Omega_0$. This result, together with the Brock and Weinstock inequalities, respectively, allows us to prove two isoperimetric inequalities for small holes.