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.
Keywords: metamaterials, interface problem, T-coercivity
@article{M2AN_2012__46_6_1363_0,
author = {Bonnet-Ben Dhia, Anne-Sophie and Chesnel, Lucas and Ciarlet, Patrick},
title = {$T$-coercivity for scalar interface problems between dielectrics and metamaterials},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1363--1387},
year = {2012},
publisher = {EDP Sciences},
volume = {46},
number = {6},
doi = {10.1051/m2an/2012006},
zbl = {1276.78008},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2012006/}
}
TY - JOUR AU - Bonnet-Ben Dhia, Anne-Sophie AU - Chesnel, Lucas AU - Ciarlet, Patrick TI - $T$-coercivity for scalar interface problems between dielectrics and metamaterials JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1363 EP - 1387 VL - 46 IS - 6 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2012006/ DO - 10.1051/m2an/2012006 LA - en ID - M2AN_2012__46_6_1363_0 ER -
%0 Journal Article %A Bonnet-Ben Dhia, Anne-Sophie %A Chesnel, Lucas %A Ciarlet, Patrick %T $T$-coercivity for scalar interface problems between dielectrics and metamaterials %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1363-1387 %V 46 %N 6 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2012006/ %R 10.1051/m2an/2012006 %G en %F M2AN_2012__46_6_1363_0
Bonnet-Ben Dhia, Anne-Sophie; Chesnel, Lucas; Ciarlet, Patrick. $T$-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1363-1387. doi: 10.1051/m2an/2012006
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