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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Conductivity interface problems. Part I: Small perturbations of an interface
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by Habib Ammari, Hyeonbae Kang, Mikyoung Lim and Habib Zribi PDF
Trans. Amer. Math. Soc. 362 (2010), 2435-2449 Request permission

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
  • MR Author ID: 353050
  • 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
  • MR Author ID: 268781
  • 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
  • MR Author ID: 689036
  • 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
  • Received by editor(s): January 13, 2006
  • Received by editor(s) in revised form: January 27, 2008
  • Published electronically: December 16, 2009
  • © Copyright 2009 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 2435-2449
  • MSC (2000): Primary 35B30
  • DOI: https://doi.org/10.1090/S0002-9947-09-04842-9
  • MathSciNet review: 2584606