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Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients

Renata Bunoiu 1 Lucas Chesnel 2, 3 Karim Ramdani 4, 1 Mahran Rihani 5, 2 
2 DeFI - Shape reconstruction and identification
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
3 IDEFIX - Inversion of Differential Equations For Imaging and physiX
Inria Saclay - Ile de France, X - École polytechnique, EDF - EDF
5 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : In this work, we are interested in the homogenization of time-harmonic Maxwell's equations in a composite medium with periodically distributed small inclusions of a negative material. Here a negative material is a material modelled by negative permittivity and permeability. Due to the sign-changing coefficients in the equations, it is not straightforward to obtain uniform energy estimates to apply the usual homogenization techniques. The goal of this article is to explain how to proceed in this context. The analysis of Maxwell's equations is based on a precise study of two associated scalar problems: one involving the sign-changing permittivity with Dirichlet boundary conditions, another involving the sign-changing permeability with Neumann boundary conditions. For both problems, we obtain a criterion on the physical parameters ensuring uniform invertibility of the corresponding operators as the size of the inclusions tends to zero. In the process, we explain the link existing with the so-called Neumann-Poincaré operator, complementing the existing literature on this topic. Then we use the results obtained for the scalar problems to derive uniform energy estimates for Maxwell's system. At this stage, an additional difficulty comes from the fact that Maxwell's equations are also sign-indefinite due to the term involving the frequency. To cope with it, we establish some sort of uniform compactness result.
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https://hal.inria.fr/hal-02421312
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Submitted on : Wednesday, May 11, 2022 - 1:14:07 PM
Last modification on : Wednesday, August 24, 2022 - 9:40:06 AM

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Renata Bunoiu, Lucas Chesnel, Karim Ramdani, Mahran Rihani. Homogenization of Maxwell's equations and related scalar problems with sign-changing coefficients. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2021, 30 (5), pp.1075-1119. ⟨10.5802/afst.1694⟩. ⟨hal-02421312v3⟩

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