Conductivity interface problems. Part I: Small perturbations of an interface

Authors:
Habib Ammari, Hyeonbae Kang, Mikyoung Lim and Habib Zribi

Journal:
Trans. Amer. Math. Soc. **362** (2010), 2435-2449

MSC (2000):
Primary 35B30

DOI:
https://doi.org/10.1090/S0002-9947-09-04842-9

Published electronically:
December 16, 2009

MathSciNet review:
2584606

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We derive high-order terms in the asymptotic expansions of boundary perturbations of steady-state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion with -boundary. Our derivation is rigorous and based on layer potential techniques. The asymptotic expansion in this paper is valid for -perturbations and inclusions with extreme conductivities. It extends those already derived for small volume conductivity inclusions and leads us to very effective algorithms for determining lower-order Fourier coefficients of the shape perturbation of the inclusion based on boundary measurements. We perform some numerical experiments using the algorithm to test its effectiveness.

**1.**H. Ammari and H. Kang,

High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter,

SIAM J. Math. Anal., 34 (2003), 1152-1166. MR**2001663 (2004e:35027)****2.**-,*Reconstruction of Small Inhomogeneities from Boundary Measurements,*Lecture Notes in Mathematics, Vol. 1846, Springer-Verlag, Berlin, 2004. MR**2168949 (2006k:35295)****3.**-,*Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory*, Applied Mathematical Sciences, Vol. 162, Springer-Verlag, New York, 2007. MR**2327884 (2009f:35339)****4.**Y. Capdeboscq and M.S. Vogelius,

A general representation formula for the boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction,

Math. Modelling Num. Anal., 37 (2003), 159-173. MR**1972656 (2004b:35334)****5.**D.J. Cedio-Fengya, S. Moskow, and M. Vogelius,

Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction,

Inverse Problems, 14 (1998), 553-595. MR**1629995 (99d:78011)****6.**T. Chan and X.-C. Tai, Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients, J. Comput. Phys., 193 (2003), 40-66. MR**2022688 (2004j:65170)****7.**R.R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur bourné sur pour les courbes lipschitziennes, Ann. Math., 116 (1982), 361-387. MR**672839 (84m:42027)****8.**E.B. Fabes, M. Jodeit, and N.M. Riviére,

Potential techniques for boundary value problems on domains,

Acta Math., 141 (1978), 165-186. MR**501367 (80b:31006)****9.**E. Fabes, H. Kang, and J.K. Seo,

Inverse conductivity problem: error estimates and approximate identification for perturbed disks,

SIAM J. Math. Anal., 30 (4) (1999), 699-720. MR**1684722 (2000d:86015)****10.**A. Friedman and M. Vogelius,

Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence,

Arch. Rat. Mech. Anal., 105 (1989), 563-579. MR**973245 (90c:35198)****11.**F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems, 14 (1998), 67-82. MR**1607628 (99b:35210)****12.**K. Ito, K. Kunish, and Z. Li, Level-set function approach to an inverse interface problem, Inverse Problems, 17 (2001), 1225-1242. MR**1862188 (2002h:35335)****13.**H. Kang and J.K. Seo,

Layer potential technique for the inverse conductivity problem,

Inverse Problems, 12 (1996), 267-278. MR**1391539 (97d:35242)****14.**-, On stability of a transmission problem, Jour. Korean Math. Soc., 34 (1997), 695-706. MR**1466611 (98g:35056)****15.**-,

Recent progress in the inverse conductivity problem with single measurement,

in*Inverse Problems and Related Fields*, CRC Press, 2000, 69-80. MR**1761339 (2001f:35427)****16.**O. Kwon, J.K. Seo, and J.R. Yoon,

A real-time algorithm for the location search of discontinuous conductivities with one measurement,

Commun. Pure Appl. Math., LV (2002), 1-29. MR**1857878 (2002g:78026)****17.**C.F. Tolmasky and A. Wiegmann, Recovery of small perturbations of an interface for an elliptic inverse problem via linearization, Inverse Problems, 15 (1999), 465-487. MR**1684468 (2000g:65111)****18.**G.C. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611. MR**769382 (86e:35038)****19.**M. Vogelius and D. Volkov,

Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities,

Math. Model. Numer. Anal., 34 (2000), 723-748. MR**1784483 (2001f:78024)**

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Additional Information

**Habib Ammari**

Affiliation:
Centre de Mathématiques Appliquées, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Cedex, France

Email:
ammari@cmapx.polytechnique.fr

**Hyeonbae Kang**

Affiliation:
Department of Mathematical Sciences and RIM, Seoul National University, Seoul 151-747, Korea

Address at time of publication:
Department of Mathematics, Inha University, Incheon 402-751, Korea

Email:
hkang@math.snu.ac.kr, hbkang@inha.ac.kr

**Mikyoung Lim**

Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523

Address at time of publication:
Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, 335 Gwahangno (373-1 Gueseong-dong), Yuseong-gu, Daejeon 305-701, Korea

Email:
lim@math.colostate.edu, mklim@kaist.ac.kr

**Habib Zribi**

Affiliation:
Centre de Mathématiques Appliquées, CNRS UMR 7641 and Ecole Polytechnique, 91128 Palaiseau Cedex, France

Address at time of publication:
Department of Mathematical Sciences, Korean Advanced Institute of Science and Technology, 335 Gwahangno (373-1 Gueseong-dong), Yuseong-gu, Daejeon 305-701, Korea

Email:
zribi@cmapx.polytechnique.fr

DOI:
https://doi.org/10.1090/S0002-9947-09-04842-9

Keywords:
Small perturbations,
interface problem,
full asymptotic expansions,
boundary integral method

Received by editor(s):
January 13, 2006

Received by editor(s) in revised form:
January 27, 2008

Published electronically:
December 16, 2009

Article copyright:
© Copyright 2009
American Mathematical Society