Extending representation formulas 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 $\Omega $, we study the perturbation incurred by the voltage potential when the conductivity is modified in a set of small measure. We consider $(\gamma _{n})_{n\,\in \,\mathbb{N}}$, a sequence of per...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2022-02-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.273/ |
| _version_ | 1797651443311706112 |
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| author | Capdeboscq, Yves Ong, Shaun Chen Yang |
| author_facet | Capdeboscq, Yves Ong, Shaun Chen Yang |
| author_sort | Capdeboscq, Yves |
| collection | DOAJ |
| description | 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 $(\gamma _{n})_{n\,\in \,\mathbb{N}}$, 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 $(\gamma _{n}-\gamma _{0})\gamma _{n}^{-1}(\gamma _{n}-\gamma _{0})\rightarrow 0$ in $L^{1}(\Omega )$. 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. |
| first_indexed | 2024-03-11T16:15:52Z |
| format | Article |
| id | doaj.art-0f5482b508c44098a44edfc1e0cb0555 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:15:52Z |
| publishDate | 2022-02-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-0f5482b508c44098a44edfc1e0cb05552023-10-24T14:19:41ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-02-01360G212715010.5802/crmath.27310.5802/crmath.273Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneitiesCapdeboscq, Yves0https://orcid.org/0000-0003-1271-867XOng, Shaun Chen Yang1Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions (LJLL), F-75006 Paris, FranceMathematical Institute, University of Oxford, OX2 6GG, UKImposing 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 $(\gamma _{n})_{n\,\in \,\mathbb{N}}$, 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 $(\gamma _{n}-\gamma _{0})\gamma _{n}^{-1}(\gamma _{n}-\gamma _{0})\rightarrow 0$ in $L^{1}(\Omega )$. 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.273/ |
| spellingShingle | Capdeboscq, Yves Ong, Shaun Chen Yang Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities Comptes Rendus. Mathématique |
| title | Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities |
| title_full | Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities |
| title_fullStr | Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities |
| title_full_unstemmed | Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities |
| title_short | Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities |
| title_sort | extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.273/ |
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