Remarks on the reinforcement of the spectrum of an elliptic problem with Robin boundary condition

Abstract

We investigate the spectral properties of a differential elliptic operator on $H^1(\bar{\Omega}\cup \Sigma)$, where $\Omega$ is a smooth domain surrounded by a layer $\Sigma$. The thickness of the layer is given by $\varepsilon h$, where $h$ is a positive function defined on the boundary $\partial \Omega$ and $\varepsilon$ is the ellipticity constant of the operator in $\Sigma$. We prove that, in the limit for $\varepsilon$ going to $0$, the spectrum converges to the spectrum of a differential elliptic operator in $H^1(\Omega)$, and we investigate a first-order asymptotic development.