T-coercivity for scalar interface problems between dielectrics and metamaterials
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, p. 1363-1387

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.

DOI : https://doi.org/10.1051/m2an/2012006
Classification:  35Q60,  35Q61,  35J20
Keywords: metamaterials, interface problem, T-coercivity
@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},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {6},
     year = {2012},
     pages = {1363-1387},
     doi = {10.1051/m2an/2012006},
     zbl = {1276.78008},
     language = {en},
     url = {http://www.numdam.org/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, Volume 46 (2012) no. 6, pp. 1363-1387. doi : 10.1051/m2an/2012006. http://www.numdam.org/item/M2AN_2012__46_6_1363_0/

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