T-coercivity for scalar interface problems between dielectrics and metamaterials

Anne-Sophie Bonnet-Ben Dhia; Lucas Chesnel; Patrick Ciarlet

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 6, page 1363-1387
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Dhia, Anne-Sophie Bonnet-Ben, Chesnel, Lucas, and Ciarlet, Patrick. "T-coercivity for scalar interface problems between dielectrics and metamaterials." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1363-1387. <http://eudml.org/doc/222107>.

@article{Dhia2012,
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.},
author = {Dhia, Anne-Sophie Bonnet-Ben, Chesnel, Lucas, Ciarlet, Patrick},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Metamaterials; interface problem; T-coercivity; dielectrics; metamaterials; Dirichlet boundary condition; coercivity; wave transmission},
language = {eng},
month = {4},
number = {6},
pages = {1363-1387},
publisher = {EDP Sciences},
title = {T-coercivity for scalar interface problems between dielectrics and metamaterials},
url = {http://eudml.org/doc/222107},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Dhia, Anne-Sophie Bonnet-Ben
AU - Chesnel, Lucas
AU - Ciarlet, Patrick
TI - T-coercivity for scalar interface problems between dielectrics and metamaterials
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/4//
PB - EDP Sciences
VL - 46
IS - 6
SP - 1363
EP - 1387
AB - 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.
LA - eng
KW - Metamaterials; interface problem; T-coercivity; dielectrics; metamaterials; Dirichlet boundary condition; coercivity; wave transmission
UR - http://eudml.org/doc/222107
ER -

References

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