×

T-coercivity for the Maxwell problem with sign-changing coefficients. (English) Zbl 1297.35229

Summary: In this paper, we study the time-harmonic Maxwell problem with sign-changing permittivity and/or permeability, set in a domain of \(\mathbb{R}^{3}\). We prove, using the T-coercivity approach, that the well-posedness of the two canonically associated scalar problems, with Dirichlet and Neumann boundary conditions, implies the well-posedness of the Maxwell problem. This allows us to give simple and sharp criteria, obtained in the study of the scalar cases, to ensure that the Maxwell transmission problem between a classical dielectric material and a negative metamaterial is well-posed.

MSC:

35Q61 Maxwell equations
35A15 Variational methods applied to PDEs
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
78A25 Electromagnetic theory (general)
78A48 Composite media; random media in optics and electromagnetic theory
78M30 Variational methods applied to problems in optics and electromagnetic theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1002/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B · Zbl 0914.35094
[2] DOI: 10.1007/BF02165003 · Zbl 0214.42001
[3] DOI: 10.1051/m2an/2012006 · Zbl 1276.78008
[4] Bonnet-Ben Dhia , A.S. Chesnel , L. , Ciarlet Jr., P. ( 2012 ).T-coercivity for the Maxwell problem with sign-changing coefficients, HAL – CCSD. Available at: http://hal.archives-ouvertes.fr/docs/00/76/22/75/PDF/BoCC12aHal.pdf (accessed 3 March 2014) . · Zbl 1297.35229
[5] Bonnet-Ben Dhia , A.S. Chesnel , L. Ciarlet , P. Jr. Two-dimensional Maxwell’s equations with sign-changing coefficients.Appl. Numer. Math.In press . · Zbl 1291.78018
[6] DOI: 10.1142/S0218202513500188 · Zbl 1283.35135
[7] DOI: 10.1016/j.cam.2006.01.046 · Zbl 1136.78003
[8] DOI: 10.1142/S0218202508003145 · Zbl 1173.35119
[9] Bonnet-Ben Dhia A.-S., J. Comp. Appl. Math. 234:1912–1919. Corrigendum: J. Comp. Appl. Math. 234 pp 2616– (2010)
[10] DOI: 10.1016/S0764-4442(99)80241-9 · Zbl 0932.35153
[11] DOI: 10.1142/2938
[12] Chesnel , L. ( 2012 ). Investigation of some transmission problems with sign-changing coefficients. Application to metamaterials. PhD thesis, École Polytechnique, Palaiseau, France .
[13] DOI: 10.1137/100810903 · Zbl 1250.78007
[14] DOI: 10.1007/s00211-012-0510-8 · Zbl 1271.65137
[15] DOI: 10.1016/j.cam.2012.09.033 · Zbl 1260.78007
[16] DOI: 10.1016/j.cam.2009.08.033 · Zbl 1192.78018
[17] DOI: 10.1016/j.crma.2012.01.017 · Zbl 1247.78011
[18] DOI: 10.1016/0022-247X(85)90118-0 · Zbl 0597.35021
[19] Dauge , M. Texier , B. ( 1997 ). Problèmes de transmission non coercifs dans des polygones [Non-coercive transmission problems in polygons]. Technical Report 97–27. Rennes, France: Université de Rennes 1, IRMAR. Available at: http://hal.archives-ouvertes.fr/docs/00/56/23/29/PDF/BenjaminT_arxiv.pdf (accessed 3 March 2014) .
[20] DOI: 10.1109/LAWP.2002.802576
[21] DOI: 10.1142/S0218202509004121 · Zbl 1205.78058
[22] DOI: 10.1007/978-3-642-61623-5 · Zbl 0585.65077
[23] DOI: 10.1137/S0036141094271259 · Zbl 0860.35129
[24] Koné , E.H. ( 2010 ). Équations intégrales volumiques pour la diffraction d’ondes électromagnétiques par un corps diélectrique [Volume Integral Equations for Electromagnetic Scattering by a Dielectric Body]. PhD thesis, Université Rennes 1, Rennes, France .
[25] DOI: 10.1002/mma.2585 · Zbl 1267.78027
[26] DOI: 10.1364/JOSAA.21.000122
[27] McLean W., Strongly Elliptic Systems and Boundary Integral Equations (2000) · Zbl 0948.35001
[28] DOI: 10.1093/acprof:oso/9780198508885.001.0001 · Zbl 1024.78009
[29] DOI: 10.1016/j.cam.2011.03.028 · Zbl 1220.65167
[30] DOI: 10.1006/jmaa.1995.1431 · Zbl 0848.35148
[31] DOI: 10.1109/TAP.2008.916955 · Zbl 1369.78688
[32] DOI: 10.1103/PhysRevLett.85.3966
[33] DOI: 10.1049/iet-smt:20060123
[34] Ramdani , K. ( 1999 ). Lignes supraconductrices: analyse mathématique et numérique [Superconducting transmission lines: Mathematical and numerical analysis]. PhD thesis, Université Paris 6, Paris, France .
[35] DOI: 10.1002/jnm.602 · Zbl 1087.78004
[36] DOI: 10.1070/PU1968v010n04ABEH003699
[37] DOI: 10.1016/j.metmat.2008.07.005
[38] DOI: 10.1002/mma.1670020103 · Zbl 0432.35032
[39] DOI: 10.1017/CBO9781139171755
[40] Zwölf , C.M. ( 2008 ). Méthodes variationnelles pour la modélisation des problèmes de transmission d’onde électromagnétique entre diélectrique et méta-matériau [Variational methods for the modeling of electromagnetic waves transmission problems between a dielectric and a metamaterial]. PhD thesis, Université Versailles-Saint-Quentin en Yvelines, Versailles, France .
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.