Asymptotics for Steady State Voltage Potentials in a Bidimensional Highly Contrasted Medium

We study the behavior of steady state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively $\alpha$) surrounded by a thin membrane of thickness $h$ and of complex permittivity $\alpha$ (respectively 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady state voltage potentials at any order as $h$ tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter $\alpha$ is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so-called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of $h$ and $\alpha$ with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviors of the potentials.