Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis

We consider an electrically conducting medium with conductivity inhomogeneities in the form of sheets of thickness 2ε. In this setup we provide a rigorous derivation of the leading terms in the asymptotic expansion of the steady state boundary voltage potentials, as ε→0. In the two-dimensional case our derivation confirms the results, heuristically obtained in E. Beretta et al., Z. Angew. Math. Phys. 52 (2001) 543–572].