Title: Conductivity interface problems. Part I: Small perturbations of an interface
Abstract: 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathcal C}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-boundary. Our derivation is rigorous and based on layer potential techniques. The asymptotic expansion in this paper is valid for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathcal C}^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-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.