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 \citep{ capdeboscq-vogelius-03a} can be extended to unbounded sequencesof matrix valued conductivities.
Categories: math.AP