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 \( \Omega _0 \subset \mathbb {R}^n \) , \( n \ge 2 \) , 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 \( r \rightarrow 0^+ \) . 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.
