Estimates for the first and second Steklov-Dirichlet eigenvalues

Abstract

In this paper, we deal with the Steklov-Dirichlet eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $\hat{I}{\text{c◯}}_r=\hat{I}{\text{c◯}}_0\setminus {\overline{B}}_r$, where $\hat{I}{\text{c◯}}_0\subset {\mathbb{R}}^n$, $n\ge 2$, 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 ${\overline{B}}_r\subset \hat{I}{\text{c◯}}_0$. In the first part of the paper, we focus on the first Steklov-Dirichlet eigenvalue $\ddot{I}{ƒ}_1(\hat{I}{\text{c◯}}_r)$ 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 $\ddot{I}{ƒ}_1(\hat{I}{\text{c◯}}_r)$ when $r$ is small enough, under a measure constraint. The second part is focused on the second Steklov-Dirichlet eigenvalue $\ddot{I}{ƒ}_2(\hat{I}{\text{c◯}}_r)$. We prove that it converges to the first non-trivial Steklov eigenvalue $\overline{\ddot{I}}{ƒ}_1(\hat{I}{\text{c◯}}_0)$ of the non-perforated domain $\hat{I}{\text{c◯}}_0$. This result, together with the Brock and Weinstock inequalities, respectively, allows us to prove two isoperimetric inequalities for small holes.

20 pages