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 $left(gamma_{n}right)_{ninmathbb{N}}$, 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 citep{capdeboscq-vogelius-03a} can be extended to unbounded sequencesof matrix valued conductivities.
Download