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HOMOGENIZATION OF THE SYSTEM OF HIGH-CONTRAST MAXWELL EQUATIONS

Part of: Elliptic equations and systems Partial differential operators

Published online by Cambridge University Press:  27 February 2015

Kirill Cherednichenko  and
Shane Cooper
Kirill Cherednichenko
Affiliation:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K. email K.Cherednichenko@bath.ac.uk
Shane Cooper
Affiliation:
Laboratoire de Mécanique et Génie Civil de Montpellier, 860 Rue de Saint-Priest, 34095 Montpellier, France email salcoops@gmail.com

Abstract

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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.

MSC classification

Primary: 35J15: Second-order elliptic equations 35J70: Degenerate elliptic equations 47F05: Partial differential operators (should also be assigned at least one other classification number in section 47)
Type
Research Article
Information
Mathematika , Volume 61 , Issue 2 , May 2015 , pp. 475 - 500
Creative Commons
Creative Common License - CCCreative Common License - BY
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.
Copyright
Copyright © University College London 2015

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