On a Steklov-Robin eigenvalue problem

TL;DR

This paper investigates the properties and asymptotic behavior of the first Steklov-Robin eigenvalue for the Laplacian in annular domains, analyzing how it varies with boundary conditions and parameters.

Contribution

It provides new insights into the eigenvalue's behavior under boundary condition variations and as the Robin parameter tends to infinity.

Findings

Eigenvalue behavior with varying boundary conditions

Asymptotic analysis as Robin parameter increases

Dependence of eigenvalues on inner radius and boundary functions

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 $\Omega_0\subset\mathbb{R}^n$, $n\geq 2$, 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.