A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity

Abstract

We revisit the asymptotic formulas originally derived in [D.J. Cedio-Fengya, S. Moskow, M.S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998) 553–595; A. Friedman, M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Ration. Mech. Anal. 105 (1989) 299–326]. These formulas concern the perturbation in the voltage potential caused by the presence of diametrically small conductivity inhomogeneities. We significantly extend the validity of the previously derived formulas, by showing that they are asymptotically correct, uniformly with respect to the conductivity of the inhomogeneities. We also extend the earlier formulas by allowing the conductivities of the inhomogeneities to be completely arbitrary $L^{\infty }$, positive definite, symmetric matrix-valued functions. We briefly discuss the relevance of the uniform asymptotic validity, and the admission of arbitrary anisotropically conducting inhomogeneities, as far as applications of the perturbation formulas to “approximate cloaking” are concerned.