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T-coercivity for scalar interface problems between dielectrics and metamaterials

Anne-Sophie Bonnet-Ben Dhia 1 Lucas Chesnel 1 Patrick Ciarlet 1, *
* Corresponding author
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
CNRS - Centre National de la Recherche Scientifique : UMR7231, UMA - Unité de Mathématiques Appliquées, Inria Saclay - Ile de France
Abstract : 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 R^d, 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.
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Submitted on : Wednesday, May 16, 2012 - 12:04:52 PM
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Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Patrick Ciarlet. T-coercivity for scalar interface problems between dielectrics and metamaterials. ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2012, 46, pp.1363-1387. ⟨10.1051/m2an/2012006⟩. ⟨hal-00564312v2⟩



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