Conductivity interface problems. Part I: Small perturbations of an interface

We derive high-order terms in the asymptotic expansions of boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion with ${\mathcal C}^2$-boundary. Our derivation is rigorous and based on layer potential techniques. The asymptotic expansion in this paper is valid for ${\mathcal C}^1$-perturbations and inclusions with extreme conductivities. It extends those already derived for small volume conductivity inclusions and leads us to very effective algorithms for determining lower-order Fourier coefficients of the shape perturbation of the inclusion based on boundary measurements. We perform some numerical experiments using the algorithm to test its effectiveness.