Partial differential equations/Mathematical physics
Stability of electromagnetic cavities perturbed by small perfectly conducting inclusions
[Stabilité des cavités électromagnétiques perturbées par des petites inclusions parfaitement conductrices]

Présenté par : the Editorial Board

Claeys, Xavier1, 2, 3
1 Sorbonne Universités, UPMC Université Paris-6, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
2 CNRS, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
3 INRIA Paris–Rocquencourt, EPI Alpines, Domaine de Voluceau, BP105, 78153 Le Chesnay cedex, France
Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 139-142.

Dans cette note, nous considérons un problème de propagation d'ondes électromagnétiques en régime harmonique dans une cavité bornée, dans le cas où la cavité contient de petites inclusions parfaitement conductrices. Nous montrons que la solution de ce problème dépend continuement des données de manière uniforme vis-à-vis de la taille des inclusions.

In this note, we consider an electromagnetic wave propagation problem in harmonic regime in a bounded cavity, in the case where the medium of propagation contains small perfectly conducting inclusions. We prove that the solution to this problem depends continuously on the data in a uniform manner with respect to the size of the inclusions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.10.009
Claeys, Xavier 1, 2, 3

1 Sorbonne Universités, UPMC Université Paris-6, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
2 CNRS, UMR 7598, Laboratoire Jacques-Louis-Lions, 75005 Paris, France
3 INRIA Paris–Rocquencourt, EPI Alpines, Domaine de Voluceau, BP105, 78153 Le Chesnay cedex, France 
@article{CRMATH_2015__353_2_139_0,
     author = {Claeys, Xavier},
     title = {Stability of electromagnetic cavities perturbed by small perfectly conducting inclusions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {139--142},
     publisher = {Elsevier},
     volume = {353},
     number = {2},
     year = {2015},
     doi = {10.1016/j.crma.2014.10.009},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2014.10.009/}
}
TY  - JOUR
AU  - Claeys, Xavier
TI  - Stability of electromagnetic cavities perturbed by small perfectly conducting inclusions
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 139
EP  - 142
VL  - 353
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2014.10.009/
DO  - 10.1016/j.crma.2014.10.009
LA  - en
ID  - CRMATH_2015__353_2_139_0
ER  - 
%0 Journal Article
%A Claeys, Xavier
%T Stability of electromagnetic cavities perturbed by small perfectly conducting inclusions
%J Comptes Rendus. Mathématique
%D 2015
%P 139-142
%V 353
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2014.10.009/
%R 10.1016/j.crma.2014.10.009
%G en
%F CRMATH_2015__353_2_139_0
Claeys, Xavier. Stability of electromagnetic cavities perturbed by small perfectly conducting inclusions. Comptes Rendus. Mathématique, Tome 353 (2015) no. 2, pp. 139-142. doi : 10.1016/j.crma.2014.10.009. http://www.numdam.org/articles/10.1016/j.crma.2014.10.009/

[1] Ammari, H.; Volkov, D. Asymptotic formulas for perturbations in the eigenfrequencies of the full Maxwell equations due to the presence of imperfections of small diameter, Asymptot. Anal., Volume 30 (2002) no. 3–4, pp. 331-350

[2] 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. (9), Volume 80 (2001) no. 8, pp. 769-814

[3] Daveau, C.; Khelifi, A. Asymptotic behaviour of the energy for electromagnetic systems in the presence of small inhomogeneities, Appl. Anal., Volume 91 (2012) no. 5, pp. 857-877

[4] Griesmaier, R. An asymptotic factorization method for inverse electromagnetic scattering in layered media, SIAM J. Appl. Math., Volume 68 (2008) no. 5, pp. 1378-1403

[5] Griesmaier, R. Detection of small buried objects: asymptotic factorization and MUSIC, Johannes Gutenberg-Universität, Mainz, Germany, 2008 (PhD thesis)

[6] Griesmaier, R. A general perturbation formula for electromagnetic fields in presence of low volume scatterers, ESAIM: Math. Model. Numer. Anal., Volume 45 (2011) no. 6, pp. 1193-1218

[7] Hiptmair, R. Finite elements in computational electromagnetism, Acta Numer., Volume 11 (2002), pp. 237-339

[8] Il'in, A.M. Study of the asymptotic behavior of the solution of an elliptic boundary value problem in a domain with a small hole, Tr. Semin. Im. I.G. Petrovskogo (1981) no. 6, pp. 57-82

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

[10] Maz'ya, V.G.; Nazarov, S.A. Asymptotic behavior of energy integrals under small perturbations of the boundary near corner and conic points, Tr. Mosk. Mat. Obŝ., Volume 50 (1987), pp. 79-129 (261)

[11] Maz'ya, V.G.; Nazarov, S.A.; Plamenevskiĭ, B.A. Asymptotic expansions of eigenvalues of boundary value problems for the Laplace operator in domains with small openings, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 48 (1984) no. 2, pp. 347-371

[12] Maz'ya, V.; Nazarov, S.; Plamenevskij, B. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. I, Operator Theory: Advances and Applications, vol. 111, Birkhäuser Verlag, Basel, 2000

[13] Nazarov, S.A. Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions, Proceedings of the St. Petersburg Mathematical Society, Vol. V, Amer. Math. Soc. Transl. Ser. 2, vol. 193, Amer. Math. Soc., Providence, RI, 1999, pp. 77-125

[14] Nazarov, S.A.; Sokołowski, J. Self-adjoint extensions for the Neumann Laplacian and applications, Acta Math. Sin. Engl. Ser., Volume 22 (2006) no. 3, pp. 879-906

[15] Picard, R. Randwertaufgaben in der verallgemeinerten Potentialtheorie, Math. Methods Appl. Sci., Volume 3 (1981) no. 2, pp. 218-228

[16] Taylor, M.E. Partial Differential Equations I. Basic Theory, Applied Mathematical Sciences, vol. 115, Springer, New York, 2011

[17] Weber, Ch. A local compactness theorem for Maxwell's equations, Math. Methods Appl. Sci., Volume 2 (1980) no. 1, pp. 12-25

Cité par Sources :