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

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