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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 ,
Lucas Chesnel  and
Patrick Ciarlet Jr.
Anne-Sophie Bonnet-Ben Dhia
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France. [email protected]; [email protected]; [email protected]
Lucas Chesnel
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France. [email protected]; [email protected]; [email protected]
Patrick Ciarlet Jr.
Affiliation:
Laboratoire POEMS, UMR 7231 CNRS/ENSTA/INRIA, ENSTA ParisTech, 32, boulevard Victor, 75739 Paris Cedex 15, France. [email protected]; [email protected]; [email protected]
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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.

Keywords

Metamaterialsinterface problemT-coercivity
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012 , pp. 1363 - 1387
Copyright
© EDP Sciences, SMAI, 2012

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References

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