Mathematical analysis/Partial differential equations
Electromagnetic scattering by periodic structures with sign-changing coefficients
Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 893-898.

We analyze the well-posedness of a scattering problem of time-harmonic electromagnetic waves by periodic structures with sign-changing coefficients. Transmission problems for Maxwell's equations with sign-changing coefficients in bounded domains have been recently studied by Bonnet-Ben Dhia and co-workers in the so-called T-coercivity framework. In this article, we generalize such a framework for periodic scattering problems relying on an integral equation approach. The periodic scattering problem is formulated by a hypersingular integral equation of Lipmann–Schwinger type. We prove that the integral equation satisfies a Gårding-type estimate, which allows us to establish the well-posedness of the problem in the sense of Fredholm.

Nous analysons le caractère bien posé du problème de diffraction d'ondes électromagnétiques par des structures périodiques dont les coefficients diélectriques changent de signe. Le problème de diffraction pour les équations de Maxwell avec des coefficients qui changent de signe a été récemment étudié par Bonnet-Ben Dhia et al. en utilisant le concept de la T-coercivité. Dans cette note, nous étendons cette étude à la diffraction par un réseau périodique en se basant sur une formulation intégrale volumique du problème. Le problème de diffraction est d'abord écrit sous la forme d'une équation de type Lippmann–Schwinger avec un noyau hyper-singulier. Nous montrons ensuite que la solution de cette équation satisfait une estimation a priori de type Gårding, ce qui nous permet de conclure sur le caractère bien posé du problème au sens de Fredholm.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.07.004
Nguyen, Dinh-Liem 1; Nguyen, Thi-Phong 2

1 Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA
2 DEFI, INRIA Saclay Île-de-France, École polytechnique, route de Saclay, 91128 Palaiseau cedex, France
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Nguyen, Dinh-Liem; Nguyen, Thi-Phong. Electromagnetic scattering by periodic structures with sign-changing coefficients. Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 893-898. doi : 10.1016/j.crma.2015.07.004. http://www.numdam.org/articles/10.1016/j.crma.2015.07.004/

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