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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

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
Published electronically: December 16, 2009
MathSciNet review: 2584606
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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 $ {\mathcal C}^2$-boundary. Our derivation is rigorous and based on layer potential techniques. The asymptotic expansion in this paper is valid for $ {\mathcal C}^1$-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.


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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: http://dx.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