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-Poincare 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.
Highlights
Negative index materials are artificially structured composite materials whose dielectric permittivity ε and magnetic permeability μ are simultaneously negative in some frequency ranges [40]
Negative index materials are interesting from a mathematical point of view
When κε = −1, the operator Aδε is an isomorphism if and only if it is injective. As it has been observed in different works, and as we recall below, the question of the injectivity of Aδε is directly linked to the spectrum of the so-called Neumann–Poincaré operator
Summary
Negative index materials ( called left-handed materials) are artificially structured composite materials whose dielectric permittivity ε and magnetic permeability μ are simultaneously negative in some frequency ranges [40]. It is proved that under this assumption on the contrast, the solution of the problem in the composite material is well-defined for δ small enough (this is not obvious due to the loss of coercivity due to the sign-changing coefficient) and that it two-scale converges (see Definition 5.1 below) to the solution of a well-posed problem set in a homogeneous material These results have been extended in [14], through the analysis of the spectrum of the Neumann–Poincaré operator. At this stage, the sign-changing of the physical parameters does not play any role. For the reader’s convenience, the list of functional spaces used throughout the paper is collected in the Appendix
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