Asymptotic expansion of steady-state potential in a high contrast medium with a thin resistive layer

Abstract

We study the steady-state potential in a high contrast medium with a resistive thin layer. We provide asymptotic expansions of the potential at any order, whatever small the layer conductivity is. Approximate transmission conditions at any order and the corresponding variational formulation are given. We prove uniform estimates with respect to the thickness of the layer and with respect to its resistivity. The main insight consists in this expansion at any order and in the corresponding non-standard variational formulation whatever small the layer conductivity is. In particular the limit potential is not continuous across the limit surface (when the membrane thickness shrinks to zero) unlike the soft contrast case and its jump is proportional to the electric flux. Numerical simulations illustrate the theoretical results.