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

Yves Capdeboscq; Shaun Chen Yang Ong

doi:10.5802/crmath.273

Équations aux dérivées partielles

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

Yves Capdeboscq ¹ ; Shaun Chen Yang Ong ²

¹ Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France
² Mathematical Institute, University of Oxford, OX2 6GG, UK

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 127-150.

Résumé

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 ${(γ_{n})}_{n \in ℕ}$ , 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 $(γ_{n} - γ_{0}) γ_{n}^{- 1} (γ_{n} - γ_{0}) \to 0$ in $L^{1} (Ω)$ . 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.

Reçu le : 2021-03-16
Révisé le : 2021-09-18
Accepté le : 2021-09-21
Publié le : 2022-02-15

DOI : 10.5802/crmath.273

Affiliations des auteurs :

Yves Capdeboscq ¹ ; Shaun Chen Yang Ong ²

¹ Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, France
² Mathematical Institute, University of Oxford, OX2 6GG, UK

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Yves Capdeboscq and Shaun Chen Yang Ong},
     title = {Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {127--150},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.273},
     language = {en},
}

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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, Volume 360 (2022), pp. 127-150. doi : 10.5802/crmath.273. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.273/

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