Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of Rd, with d = 2,3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive + compact) framework. For that, we build some criteria, based on the geometry of the interface between the dielectric and the metamaterial. The proofs combine simple geometrical arguments with the approach of T-coercivity, introduced by the first and third authors and co-worker. Furthermore, the use of localization techniques allows us to derive well-posedness under conditions that involve the knowledge of the coefficients only near the interface. When the coefficients are piecewise constant, we establish the optimality of the criteria.
@article{M2AN_2012__46_6_1363_0,
author = {Dhia, Anne-Sophie Bonnet-Ben and Chesnel, Lucas and Ciarlet, Patrick},
title = {$T$-coercivity for scalar interface problems between dielectrics and metamaterials},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {46},
year = {2012},
pages = {1363-1387},
doi = {10.1051/m2an/2012006},
zbl = {1276.78008},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2012__46_6_1363_0}
}
Dhia, Anne-Sophie Bonnet-Ben; Chesnel, Lucas; Ciarlet, Patrick. $T$-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) pp. 1363-1387. doi : 10.1051/m2an/2012006. http://gdmltest.u-ga.fr/item/M2AN_2012__46_6_1363_0/
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