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