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HOMOGENIZATION OF THE SYSTEM OF HIGH-CONTRAST MAXWELL EQUATIONS

  • Kirill Cherednichenko (a1) and Shane Cooper (a2)
Abstract

We study the system of Maxwell equations for a periodic composite dielectric medium with components whose dielectric permittivities ${\it\epsilon}$ have a high degree of contrast between each other. We assume that the ratio between the permittivities of the components with low and high values of ${\it\epsilon}$ is of the order ${\it\eta}^{2}$, where ${\it\eta}>0$ is the period of the medium. We determine the asymptotic behaviour of the electromagnetic response of such a medium in the “homogenization limit”, as ${\it\eta}\rightarrow 0$, and derive the limit system of Maxwell equations in $\mathbb{R}^{3}$. Our results extend a number of conclusions of a paper by Zhikov [On gaps in the spectrum of some divergent elliptic operators with periodic coefficients. St. Petersburg Math. J.16(5) (2004), 719–773] to the case of the full system of Maxwell equations.

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This article is distributed with Open Access under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided that the original work is properly cited.
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G. Allaire , Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 1992, 14821518.

K. D. Cherednichenko , S. Cooper and S. Guenneau , Spectral analysis of one-dimensional high-contrast elliptic problems with periodic coefficients. Multiscale Model. Simul. 13 2015, 7298.

D. Cioranescu , A. Damlamian and G. Griso , Periodic unfolding and homogenisation. C. R. Math. Acad. Sci. Paris 335(1) 2002, 99104.

R. Dautray and J.-L. Lions , Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3 (Spectral Theory and Applications), Springer (Berlin, 1990).

A. Figotin and P. Kuchment , Spectral properties of classical waves in high-contrast periodic media. SIAM J. Appl. Math. 58 1998, 683702.

R. Hempel and K. Lienau , Spectral properties of periodic media in the large coupling limit. Comm. Partial Differential Equations 25 2000, 14451470.

V. V. Jikov , S. M. Kozlov and O. A. Oleinik , Homogenization of Differential Operators and Integral Functionals, Springer (1994).

P. Kuchment , Floquet Theory for Partial Differential Equations (Operator Theory: Advances and Applications 60), Birkhäuser (Basel, 1993).

V. G. Maz’ya , Sobolev Spaces, Springer (Berlin–Tokyo, 1985).

V. G. Maz’ya , S. A. Nazarov and B. A. Plamenevskii , Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. II (Operator Theory: Advances and Applications 112), Birkhäuser (Basel, 2000).

G. Nguetseng , A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3) 1989, 608623.

L. Parnovski , Bethe–Sommerfeld conjecture. Ann. Henri Poincaré 9(3) 2008, 457508.

S. E. Pastukhova , Asymptotic analysis of elasticity problems on thin periodic structures. Netw. Heterog. Media 4(3) 2009, 577604.

P. Russell , Photonic crystal fibers. Science 299 2003, 358362.

V. V. Zhikov , On an extension of the method of two-scale convergence and its applications. Sb. Math. 191(7) 2000, 9731014.

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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
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