On the asymptotic behavior of a diffraction problem with a thin layer

We investigate the behavior of the solution to an elliptic diffraction problem in the union of a smooth set $\Omega$ and a thin layer $\Sigma$ locally described by $\varepsilon h$, where $h$ is a positive function defined on the boundary $\partial\Omega$, and $\varepsilon$ is the ellipticity constant of the differential operator in the thin layer $\Sigma$. We study the problem in the limit for $\varepsilon$ going to zero and prove a first-order asymptotic development by $\Gamma$-convergence of the associated energy functional.