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T-coercivity for scalar interface problems between dielectrics and metamaterials

Published online by Cambridge University Press:  11 April 2012

Anne-Sophie Bonnet-Ben Dhia
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France. anne-sophie.bonnet-bendhia@ensta-paristech.fr; lucas.chesnel@ensta-paristech.fr; patrick.ciarlet@ensta-paristech.fr
Lucas Chesnel
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France. anne-sophie.bonnet-bendhia@ensta-paristech.fr; lucas.chesnel@ensta-paristech.fr; patrick.ciarlet@ensta-paristech.fr
Patrick Ciarlet Jr.
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France. anne-sophie.bonnet-bendhia@ensta-paristech.fr; lucas.chesnel@ensta-paristech.fr; patrick.ciarlet@ensta-paristech.fr
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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 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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2012

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References

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