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), 24352449
MSC (2000):
Primary 35B30
Published electronically:
December 16, 2009
MathSciNet review:
2584606
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We derive highorder terms in the asymptotic expansions of boundary perturbations of steadystate 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 lowerorder 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.
Habib
Ammari and Hyeonbae
Kang, Highorder terms in the asymptotic expansions of the
steadystate voltage potentials in the presence of conductivity
inhomogeneities of small diameter, SIAM J. Math. Anal.
34 (2003), no. 5, 1152–1166 (electronic). MR 2001663
(2004e:35027), 10.1137/S0036141001399234
 2.
Habib
Ammari and Hyeonbae
Kang, Reconstruction of small inhomogeneities from boundary
measurements, Lecture Notes in Mathematics, vol. 1846,
SpringerVerlag, Berlin, 2004. MR 2168949
(2006k:35295)
 3.
Habib
Ammari and Hyeonbae
Kang, Polarization and moment tensors, Applied Mathematical
Sciences, vol. 162, Springer, New York, 2007. With applications to
inverse problems and effective medium theory. MR 2327884
(2009f:35339)
 4.
Yves
Capdeboscq and 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. 37 (2003),
no. 1, 159–173. MR 1972656
(2004b:35334), 10.1051/m2an:2003014
 5.
D.
J. CedioFengya, S.
Moskow, and M.
S. Vogelius, Identification of conductivity imperfections of small
diameter by boundary measurements. Continuous dependence and computational
reconstruction, Inverse Problems 14 (1998),
no. 3, 553–595. MR 1629995
(99d:78011), 10.1088/02665611/14/3/011
 6.
Tony
F. Chan and XueCheng
Tai, Level set and total variation regularization for elliptic
inverse problems with discontinuous coefficients, J. Comput. Phys.
193 (2004), no. 1, 40–66. MR 2022688
(2004j:65170), 10.1016/j.jcp.2003.08.003
 7.
R.
R. Coifman, A.
McIntosh, and Y.
Meyer, L’intégrale de Cauchy définit un
opérateur borné sur 𝐿² pour les courbes
lipschitziennes, Ann. of Math. (2) 116 (1982),
no. 2, 361–387 (French). MR 672839
(84m:42027), 10.2307/2007065
 8.
E.
B. Fabes, M.
Jodeit Jr., and N.
M. Rivière, Potential techniques for boundary value problems
on 𝐶¹domains, Acta Math. 141 (1978),
no. 34, 165–186. MR 501367
(80b:31006), 10.1007/BF02545747
 9.
Eugene
Fabes, Hyeonbae
Kang, and Jin
Keun Seo, Inverse conductivity problem with one measurement: error
estimates and approximate identification for perturbed disks, SIAM J.
Math. Anal. 30 (1999), no. 4, 699–720
(electronic). MR
1684722 (2000d:86015), 10.1137/S0036141097324958
 10.
Avner
Friedman and Michael
Vogelius, Identification of small inhomogeneities of extreme
conductivity by boundary measurements: a theorem on continuous
dependence, Arch. Rational Mech. Anal. 105 (1989),
no. 4, 299–326. MR 973245
(90c:35198), 10.1007/BF00281494
 11.
Frank
Hettlich and William
Rundell, The determination of a discontinuity in a conductivity
from a single boundary measurement, Inverse Problems
14 (1998), no. 1, 67–82. MR 1607628
(99b:35210), 10.1088/02665611/14/1/008
 12.
Kazufumi
Ito, Karl
Kunisch, and Zhilin
Li, Levelset function approach to an inverse interface
problem, Inverse Problems 17 (2001), no. 5,
1225–1242. MR 1862188
(2002h:35335), 10.1088/02665611/17/5/301
 13.
Hyeonbae
Kang and Jin
Keun Seo, The layer potential technique for the inverse
conductivity problem, Inverse Problems 12 (1996),
no. 3, 267–278. MR 1391539
(97d:35242), 10.1088/02665611/12/3/007
 14.
Hyeonbae
Kang and Jin
Keun Seo, On stability of a transmission problem, J. Korean
Math. Soc. 34 (1997), no. 3, 695–706. MR 1466611
(98g:35056)
 15.
Hyeonbae
Kang and Jin
Keun Seo, Recent progress in the inverse conductivity problem with
single measurement, Inverse problems and related topics (Kobe, 1998)
Chapman & Hall/CRC Res. Notes Math., vol. 419, Chapman &
Hall/CRC, Boca Raton, FL, 2000, pp. 69–80. MR 1761339
(2001f:35427)
 16.
Ohin
Kwon, Jin
Keun Seo, and JeongRock
Yoon, A realtime algorithm for the location search of
discontinuous conductivities with one measurement, Comm. Pure Appl.
Math. 55 (2002), no. 1, 1–29. MR 1857878
(2002g:78026), 10.1002/cpa.3009
 17.
Carlos
F. Tolmasky and Andreas
Wiegmann, Recovery of small perturbations of an interface for an
elliptic inverse problem via linearization, Inverse Problems
15 (1999), no. 2, 465–487. MR 1684468
(2000g:65111), 10.1088/02665611/15/2/008
 18.
Gregory
Verchota, Layer potentials and regularity for the Dirichlet problem
for Laplace’s equation in Lipschitz domains, J. Funct. Anal.
59 (1984), no. 3, 572–611. MR 769382
(86e:35038), 10.1016/00221236(84)900661
 19.
Michael
S. Vogelius and Darko
Volkov, Asymptotic formulas for perturbations in the
electromagnetic fields due to the presence of inhomogeneities of small
diameter, M2AN Math. Model. Numer. Anal. 34 (2000),
no. 4, 723–748. MR 1784483
(2001f:78024), 10.1051/m2an:2000101
 1.
 H. Ammari and H. Kang,
Highorder terms in the asymptotic expansions of the steadystate voltage potentials in the presence of conductivity inhomogeneities of small diameter, SIAM J. Math. Anal., 34 (2003), 11521166. MR 2001663 (2004e:35027)
 2.
 , Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, Vol. 1846, SpringerVerlag, 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, SpringerVerlag, 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), 159173. MR 1972656 (2004b:35334)
 5.
 D.J. CedioFengya, S. Moskow, and M. Vogelius,
Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, 14 (1998), 553595. 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), 4066. 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), 361387. 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), 165186. 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), 699720. 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), 563579. 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), 6782. MR 1607628 (99b:35210)
 12.
 K. Ito, K. Kunish, and Z. Li, Levelset function approach to an inverse interface problem, Inverse Problems, 17 (2001), 12251242. MR 1862188 (2002h:35335)
 13.
 H. Kang and J.K. Seo,
Layer potential technique for the inverse conductivity problem, Inverse Problems, 12 (1996), 267278. MR 1391539 (97d:35242)
 14.
 , On stability of a transmission problem, Jour. Korean Math. Soc., 34 (1997), 695706. MR 1466611 (98g:35056)
 15.
 ,
Recent progress in the inverse conductivity problem with single measurement, in Inverse Problems and Related Fields, CRC Press, 2000, 6980. MR 1761339 (2001f:35427)
 16.
 O. Kwon, J.K. Seo, and J.R. Yoon,
A realtime algorithm for the location search of discontinuous conductivities with one measurement, Commun. Pure Appl. Math., LV (2002), 129. 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), 465487. 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), 572611. 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), 723748. MR 1784483 (2001f:78024)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
35B30
Retrieve articles in all journals
with MSC (2000):
35B30
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 151747, Korea
Address at time of publication:
Department of Mathematics, Inha University, Incheon 402751, 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 (3731 Gueseongdong), Yuseonggu, Daejeon 305701, 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 (3731 Gueseongdong), Yuseonggu, Daejeon 305701, Korea
Email:
zribi@cmapx.polytechnique.fr
DOI:
http://dx.doi.org/10.1090/S0002994709048429
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
