Extending Representation Formulae for Boundary Voltage Perturbations of Low Volume Fraction to Very Contrasted Conductivity Inhomogeneities

Abstract

Imposing either Dirichlet or Neumann boundary conditions on the boundary of a smooth bounded domain $\Omega$, we study the perturbation incurred by the voltage potential when the conductivity is modified in a set of small measure. We consider ${\left({\gamma }_n\right)}_{n\in \mathbb{N}}$, a sequence of perturbed conductivity matrices differing from a smooth ${\gamma }_0$ background conductivity matrix on a measurable set well within the domain, and we assume $\left({\gamma }_n-{\gamma }_0\right){\gamma }_n^{-1}\left({\gamma }_n-{\gamma }_0\right)\to 0$ in $L^1(\Omega )$. Adapting the limit measure, we show that the general representation formula introduced for bounded contrasts in \citep{capdeboscq-vogelius-03a} can be extended to unbounded sequencesof matrix valued conductivities.