Extending Representation Formulae for Boundary Voltage Perturbations of Low Volume Fraction to Very Contrasted Conductivity Inhomogeneities

TL;DR

This paper extends mathematical formulas for boundary voltage perturbations caused by small, highly contrasted inhomogeneities within a domain, accommodating unbounded conductivity sequences and broadening previous bounded contrast results.

Contribution

It generalizes existing representation formulas to include unbounded conductivity sequences with high contrast, expanding the applicability of boundary perturbation analysis.

Findings

Extended representation formula to unbounded conductivities.

Applicable to very contrasted inhomogeneities within the domain.

Maintains validity under small volume perturbations.

Abstract

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.