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Journal Articles Comptes Rendus. Mathématique Year : 2022

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

Yves Capdeboscq
Shaun Chen Yang Ong
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#### 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 this article can be extended to unbounded sequences of matrix valued conductivities.

#### Domains

Mathematics [math] Analysis of PDEs [math.AP]

### Dates and versions

hal-03171353 , version 1 (16-03-2021)
hal-03171353 , version 2 (22-02-2022)

### Licence

Attribution - CC BY 4.0

### Identifiers

• HAL Id : hal-03171353 , version 2
• DOI :

### Cite

Yves Capdeboscq, Shaun Chen Yang Ong. Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities. Comptes Rendus. Mathématique, 2022, 360, pp.127--150. ⟨10.5802/crmath.273⟩. ⟨hal-03171353v2⟩

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