On a Steklov-Robin eigenvalue problem

Abstract

In this paper we study a Steklov-Robin eigenvalue problem for the Laplacian in annular domains. More precisely, we consider $\Omega ={\Omega }_0\setminus {\overline{B}}_r$, where $B_r$ is the ball centered at the origin with radius $r>0$ and ${\Omega }_0\subset {\mathbb{R}}^n$, $n\ge 2$, is an open, bounded set with Lipschitz boundary, such that ${\overline{B}}_r\subset {\Omega }_0$. We impose a Steklov condition on the outer boundary and a Robin condition involving a positive $L^{\infty }$-function $\beta (x)$ on the inner boundary. Then, we study the first eigenvalue ${\sigma }_{\beta }(\Omega )$ and its main properties. In particular, we investigate the behaviour of ${\sigma }_{\beta }(\Omega )$ when we let vary the $L^1$-norm of $\beta$ and the radius of the inner ball. Furthermore, we study the asymptotic behaviour of the corresponding eigenfunctions when $\beta$ is a positive parameter that goes to infinity.

arXiv comment21 pages