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

Abstract

We investigate the behavior of the solution to an elliptic diffraction problem in the union of a smooth set $\hat{I}\text{c◯}$ and a thin layer $\hat{I}\pounds$ locally described by $\epsilon h$, where $h$ is a positive function defined on the boundary $\partial \hat{I}\text{c◯}$, and $\epsilon$ is the ellipticity constant of the differential operator in the thin layer $\hat{I}\pounds$. We study the problem in the limit for $\epsilon$ going to zero and prove a first-order asymptotic development by $\hat{I}“$-convergence of the associated energy functional.