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

@article{CRMATH_2022__360_G2_127_0,
     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},
}

TY  - JOUR
AU  - Yves Capdeboscq
AU  - Shaun Chen Yang Ong
TI  - Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 127
EP  - 150
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.273
LA  - en
ID  - CRMATH_2022__360_G2_127_0
ER  -

%0 Journal Article
%A Yves Capdeboscq
%A Shaun Chen Yang Ong
%T Extending representation formulas for boundary voltage perturbations of low volume fraction to very contrasted conductivity inhomogeneities
%J Comptes Rendus. Mathématique
%D 2022
%P 127-150
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.273
%G en
%F CRMATH_2022__360_G2_127_0

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/

Bibliographie
Cité par

[1] Giovanni S. Alberti; Yves Capdeboscq Lectures on Elliptic Methods for Hybrid Inverse Problems, Cours Spécialisés (Paris), 25, Société Mathématique de France, 2018 | Zbl

[2] Habib Ammari; Hyeonbae Kang Reconstruction of small inhomogeneities from boundary measurements, Lecture Notes in Mathematics, 1846, Springer, 2004 | DOI | Zbl

[3] Fabrice Bethuel; Haïm Brezis; Frédéric Hélein Ginzburg–Landau vortices, Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser, 2017 | DOI | Zbl

[4] Martin Brühl; Martin Hanke; Michael S. Vogelius A direct impedance tomography algorithm for locating small inhomogeneities, Numer. Math., Volume 93 (2003) no. 4, pp. 635-654 | DOI | MR | Zbl

[5] Yves Capdeboscq On the scattered field generated by a ball inhomogeneity of constant index, Asymptotic Anal., Volume 77 (2012) no. 3-4, pp. 197-246 | DOI | MR | Zbl

[6] Yves Capdeboscq Corrigendum: On the scattered field generated by a ball inhomogeneity of constant index, Asymptotic Anal., Volume 88 (2014) no. 3, pp. 185-186 | DOI | Zbl

[7] Yves Capdeboscq; George Leadbetter; Andrew Parker On the scattered field generated by a ball inhomogeneity of constant index in dimension three, Multi-scale and high-contrast PDE: from modelling, to mathematical analysis, to inversion. Proceedings of the conference, University of Oxford, UK, June 28 – July 1, 2011 (Habib Ammari, ed.) (Contemporary Mathematics), Volume 577, American Mathematical Society, 2012, pp. 61-80 | DOI | MR | Zbl

[8] Yves Capdeboscq; Michael S. Vogelius A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, M2AN, Math. Model. Numer. Anal., Volume 37 (2003) no. 1, pp. 159-173 | DOI | Numdam | MR | Zbl

[9] Yves Capdeboscq; Michael S. Vogelius A review of some recent work on impedance imaging for inhomogeneities of low volume fraction, Partial differential equations and inverse problems. Proceedings of the Pan-American Advanced Studies Institute on partial differential equations, nonlinear analysis and inverse problems, Santiago, Chile, January 6–18, 2003 (Carlos Conca, ed.) (Contemporary Mathematics), Volume 362, American Mathematical Society, 2004, pp. 69-87 | DOI | MR | Zbl

[10] Yves Capdeboscq; Michael S. Vogelius Pointwise polarization tensor bounds, and applications to voltage perturbations caused by thin inhomogeneities, Asymptotic Anal., Volume 50 (2006) no. 3-4, pp. 175-204 | MR | Zbl

[11] Charles Dapogny; Michael S. Vogelius Uniform asymptotic expansion of the voltage potential in the presence of thin inhomogeneities with arbitrary conductivity, Chin. Ann. Math., Ser. B, Volume 38 (2017) no. 1, pp. 293-344 | DOI | MR | Zbl

[12] Giuseppe Di Fazio $L^{p}$ -estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Unione Mat. Ital., VII. Ser., A, Volume 10 (1996) no. 2, pp. 409-420 | MR | Zbl

[13] Graeme W. Milton The theory of composites, Cambridge Monographs on Applied and Computational Mathematics, 6, Cambridge University Press, 2002 | DOI | Zbl

[14] Graeme W. Milton; Nicolae-Alexandru P. Nicorovici On the cloaking effects associated with anomalous localized resonances, Proc. R. Soc. Lond., Ser. A, Volume 462 (2006) no. 2074, pp. 3027-3059 | MR | Zbl

[15] Andrea Moiola; Euan A. Spence Acoustic transmission problems: wavenumber-explicit bounds and resonance-free regions, Math. Models Methods Appl. Sci., Volume 29 (2019) no. 2, pp. 317-354 | DOI | MR | Zbl

[16] Hoai-Minh Nguyen Invisibilité par résonance localisée anormale. Une liaison entre la résonance localisée et l’exposion de la puissance pour les milieux doublement complémentaires, C. R. Math. Acad. Sci. Paris, Volume 353 (2006) no. 1, pp. 41-46 | Zbl

[17] Hoai-Minh Nguyen Cloaking via anomalous localized resonance for doubly complementary media in the finite frequency regime, J. Anal. Math., Volume 138 (2019) no. 1, pp. 157-184 | DOI | MR | Zbl

[18] Hoai-Minh Nguyen; Michael S. Vogelius A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 26 (2009) no. 6, pp. 2283-2315 | DOI | Numdam | MR | Zbl

[19] Shaun Chen Yang Ong The Jacobian of Solutions to the Conductivity Equation and Problems arising from EIT, Ph. D. Thesis, University of Oxford, Oxford, United Kingdom (2019)

[20] Georgi Popov; Georgi Vodev Resonances near the real axis for transparent obstacles, Commun. Math. Phys., Volume 207 (1999) no. 2, pp. 411-438 | DOI | MR | Zbl

[21] Guido Stampacchia Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, Volume 15 (1965) no. 1, pp. 189-257 | DOI | Numdam | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Bounds on the volume fraction of 2-phase, 2-dimensional elastic bodies and on (stress, strain) pairs in composites

Graeme Walter Milton; Loc Hoang Nguyen

C. R. Méca (2012)

Homogenization of a convection–diffusion model with reaction in a porous medium

Grégoire Allaire; Anne-Lise Raphael

C. R. Math (2007)

A connection between topological ligaments in shape optimization and thin tubular inhomogeneities

Charles Dapogny

C. R. Math (2020)