Extending Representation Formulae 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 $Ω$, we study the perturbation incurred by the voltage potential when the conductivity is modified in a set of small measure. We consider $\left(γ_{n}\right)_{n\in\mathbb{N}}$, a sequence of perturbed conductivity matrices differing from a smooth $γ_{0}$ background conductivity matrix on a measurable set well within the domain, and we assume $\left(γ_{n}-γ_{0}\right)γ_{n}^{-1}\left(γ_{n}-γ_{0}\right)\to0$ in $L^{1}(Ω)$. 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.
math.AP↗