Extending Representation Formulae 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 $\left(\gamma_{n}\right)_{n\in\mathbb{N}}$, a sequence of...
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| creator | Capdeboscq, Yves Ong, Shaun Chen Yang |
| 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 $\left(\gamma_{n}\right)_{n\in\mathbb{N}}$, 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. |
| doi_str_mv | 10.48550/arxiv.2103.09644 |
| format | Article |
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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.</description><identifier>DOI: 10.48550/arxiv.2103.09644</identifier><language>eng</language><subject>Mathematics - Analysis of PDEs</subject><creationdate>2021-03</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,780,885</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2103.09644$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2103.09644$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Capdeboscq, Yves</creatorcontrib><creatorcontrib>Ong, Shaun Chen Yang</creatorcontrib><title>Extending Representation Formulae for Boundary Voltage Perturbations of Low Volume Fraction to Very Contrasted Conductivity Inhomogeneities</title><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 $\left(\gamma_{n}\right)_{n\in\mathbb{N}}$, 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.</description><subject>Mathematics - Analysis of PDEs</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotkE1OwzAQhbNhgQoHYIUv0OAkdpwuoWqhUiQQqrqtxvY4WErsynFKewYuTRJYzZPejzRfkjxkNGUV5_QJwsWe0zyjRUpXJWO3yc_mEtFp6xryiaeAPboI0XpHtj50QwtIjA_kxQ9OQ7iSg28jNEg-MMQhyDnaE29I7b8nc-iQbAOoeSJ6csCxtPYuBugj6knqYXTPNl7Jzn35zjfo0EaL_V1yY6Dt8f7_LpL9drNfvy3r99fd-rleQinY0gDNFKW8UKUWwChnQrOc0zwXFUCZZyilFEZolZcVZxUorTiVRq6YyaQSxSJ5_JudaRxPwXbjZ8eJynGmUvwCIKBhOA</recordid><startdate>20210317</startdate><enddate>20210317</enddate><creator>Capdeboscq, Yves</creator><creator>Ong, Shaun Chen Yang</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20210317</creationdate><title>Extending Representation Formulae for Boundary Voltage Perturbations of Low Volume Fraction to Very Contrasted Conductivity Inhomogeneities</title><author>Capdeboscq, Yves ; Ong, Shaun Chen Yang</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a674-fa01c0053c6d7a40547d42502278aa621ebbb7f7dc268548acdc50bfb94f1bc73</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Mathematics - Analysis of PDEs</topic><toplevel>online_resources</toplevel><creatorcontrib>Capdeboscq, Yves</creatorcontrib><creatorcontrib>Ong, Shaun Chen Yang</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Capdeboscq, Yves</au><au>Ong, Shaun Chen Yang</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Extending Representation Formulae for Boundary Voltage Perturbations of Low Volume Fraction to Very Contrasted Conductivity Inhomogeneities</atitle><date>2021-03-17</date><risdate>2021</risdate><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.</abstract><doi>10.48550/arxiv.2103.09644</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Analysis of PDEs |
| title | Extending Representation Formulae for Boundary Voltage Perturbations of Low Volume Fraction to Very Contrasted Conductivity Inhomogeneities |
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