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

Bunoiu, Renata; Chesnel, Lucas; Ramdani, Karim; Rihani, Mahran

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 5, pp. 1075-1119.

Dans ce travail, nous nous intéressons à l’homogénéisation des équations de Maxwell harmoniques dans un milieu composite contenant une distribution périodique de petites inclusions de matériau négatif. On désigne ici par matériau négatif un matériau décrit par une permittivité et une perméabilité négatives. En raison du changement de signe des coefficients intervenant dans les équations, il n’est pas évident d’obtenir des estimations d’énergie uniformes et d’appliquer les techniques d’homogénéisation classiques. Le but de cet article est d’indiquer comment on peut néanmoins procéder dans ce contexte. L’analyse des équations de Maxwell est basée sur une étude précise de deux problèmes scalaires : l’un faisant intervenir la permittivité changeant de signe avec des conditions aux limites de Dirichlet, et l’autre la perméabilité changeant de signe avec des conditions aux limites de Neumann. Pour chacun de ces deux problèmes, on obtient un critère portant sur les paramètres physiques garantissant l’inversibilité uniforme des opérateurs associés lorsque la taille des inclusions tend vers zéro. Incidemment, nous expliquons le lien existant avec l’opérateur de Neumann–Poincaré, complétant la littérature existant sur le sujet. Les résultats obtenus pour les problèmes scalaires sont ensuite utilisés pour obtenir des estimations d’énergie uniforme pour le système de Maxwell. A ce stade, il faut contourner une difficulté supplémentaire liée au caractère indéfini induit par le terme fréquentiel. Ceci est réalisé en établissant un résultat de type compacité uniforme.

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.

Classification : 10X99, 14A12, 11L05
Keywords: Homogenization, Maxwell’s equations, metamaterials, sign-changing coefficients, Neumann–Poincaré operator
Mots clés : Homogénéisation, équations de Maxwell, métamatériaux, coefficients changeant de signe, opérateur de Neumann–Poincaré

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