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

Capdeboscq, Yves; Ong, Shaun Chen Yang

Comptes Rendus. Mathématique, Tome 360 (2022) no. G2, pp. 127-150.

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 $({\gamma }_n-{\gamma }_0){\gamma }_n^{-1}({\gamma }_n-{\gamma }_0)\to 0$ in $L^1\left(\Omega \right)$. Adapting the limit measure, we show that the general representation formula introduced for bounded contrasts in a previous work from 2003 can be extended to unbounded sequences of matrix valued conductivities.

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