Partial Differential Equations
On the perturbation of the electromagnetic energy due to the presence of small inhomogeneities
Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 287-292.

We consider solutions to the time-harmonic Maxwell problem in R3. For such solution we propose a rigorous derivation of the asymptotic expansions in the interesting practical situation when a finite number of inhomogeneities of small diameter are embedded in the entire space. Then, we describe the behavior of the electromagnetic energy caused by the presence of these inhomogeneities.

Nous considérons des solutions des équations de Maxwell dans R3 en présence d'un nombre fini d'inhomogénéités de petits diamètres. Pour de telles solutions, nous obtenons des formules asymptotiques rigoureuses. Puis, nous décrivons le comportement de l'énergie électromagnétique.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2008.01.019
Daveau, Christian 1; Khelifi, Abdessatar 2

1 Université de Cergy-Pontoise, département de mathématique, CNRS UMR 8088, 2, avenue Adolphe-Chauvin, 95302 Cergy-Pontoise, France
2 Département de mathématiques, université des sciences de Carthage, Bizerte, 7021, Tunisia
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Daveau, Christian; Khelifi, Abdessatar. On the perturbation of the electromagnetic energy due to the presence of small inhomogeneities. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 287-292. doi : 10.1016/j.crma.2008.01.019. http://www.numdam.org/articles/10.1016/j.crma.2008.01.019/

[1] Ammari, H.; Iakovleva, E.; Lesselier, D. Two numerical methods for recovering small inclusions from the scattering amplitude at a fixed frequency, SIAM J. Sci. Comput., Volume 27 (2005), pp. 130-158

[2] Ammari, H.; Kang, H. Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, vol. 1846, Springer-Verlag, Berlin, 2004

[3] Ammari, H.; Khelifi, A. Electromagnetic scattering by small dielectric inhomogeneities, J. Math. Pures Appl., Volume 82 (2003), pp. 749-842

[4] Ammari, H.; Vogelius, M.S.; Volkov, D. Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. The full Maxwell equations, J. Math. Pures Appl., Volume 80 (2001), pp. 769-814

[5] Ammari, H.; Volkov, D. Correction of order three for the expansion of two dimensional electromagnetic fields perturbed by the presence of inhomogeneities of small diameter, J. Comput. Phys., Volume 189 (2003), pp. 371-389

[6] Cedio-Fengya, D.J.; Moskow, S.; Vogelius, M. Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems, Volume 14 (1998), pp. 553-595

[7] Cole, J.D. Perturbation Methods in Applied Mathematics, Blaisdell, Walthan, MA, 1968

[8] Friedman, A.; Vogelius, M. Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Arch. Rat. Mech. Anal., Volume 105 (1989), pp. 299-326

[9] Hazard, C.; Lenoir, M. On the solution of time-harmonic scattering problems for Maxwell's equations, SIAM J. Math. Anal., Volume 27 (1996) no. 6, pp. 1597-1630

[10] Il'in, A.M. Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Translations of Mathematical Monographs, vol. 102, American Mathematical Society, Providence, RI, 1992

[11] Movchan, A.B.; Serkov, S.K. The Pólya–Szegö matrices in asymptotic models of dilute media, Eur. J. Appl. Math., Volume 8 (1997), pp. 595-621

[12] Nédélec, J.C. Acoustic and Electromagnetic Equations. Integral Representation for Harmonic Problem, Springer-Verlag, New York, 2001

[13] Sanchez Hubert, J.; Sanchez Palencia, E. Vibration and Coupling of Continuous Systems, Springer-Verlag, 1989

[14] Vogelius, M.; Volkov, D. Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities, Math. Model. Numer. Anal., Volume 34 (2000), pp. 723-748

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