Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities

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 $\left(\gamma_{n}-\gamma_{0}\right)\gamma_{n}^{-1}\left(\gamma_{n}-\gamma_{0}\right)\to0$ in $L^{1}(\Omega)$. Adapting the limit measure, we show that the general representation formula introduced for bounded contrasts in this article can be extended to unbounded sequencesof matrix valued conductivities.