Conductivity Interface Problems. Part I: Small Perturbations of an Interface

We derive high-order terms in the asymptotic expansions of the boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion with C2-boundary. Our derivation is rigorous and based on layer potential techniques. The asymptotic expansion in this paper is valid for C1-perturbations and inclusions with extreme conductivities. It extends those already derived for small volume conductivity inclusions and is expected to lead to very effective algorithms, aimed at determining certain properties of the shape of a conductivity inclusion based on boundary measurements.