Skip to main content
×
Home

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      HOMOGENIZATION OF THE SYSTEM OF HIGH-CONTRAST MAXWELL EQUATIONS
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      HOMOGENIZATION OF THE SYSTEM OF HIGH-CONTRAST MAXWELL EQUATIONS
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      HOMOGENIZATION OF THE SYSTEM OF HIGH-CONTRAST MAXWELL EQUATIONS
      Available formats
      ×
Copyright
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.
References
Hide All
1.Adams R. A., Sobolev Spaces, Academic Press (New York–San Francisco–London, 1975).
2.Allaire G., Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 1992, 14821518.
3.Bensoussan A., Lions J.-L. and Papanicolaou G. C., Asymptotic Analysis for Periodic Structures, North-Holland (1978).
4.Berkolaiko G. and Kuchment P., Introduction to Quantum Graphs (Mathematical Surveys and Monographs 186), American Mathematical Society (Providence, RI, 2013).
5.Cherednichenko K. D., Two-scale asymptotics for non-local effects in composites with highly anisotropic fibres. Asymptot. Anal. 49 2006, 3959.
6.Cherednichenko K. D. and Cooper S., Resolvent estimates for high-contrast elliptic problems with periodic coefficients, submitted (2013).
7.Cherednichenko K. D., Cooper S. and Guenneau S., Spectral analysis of one-dimensional high-contrast elliptic problems with periodic coefficients. Multiscale Model. Simul. 13 2015, 7298.
8.Cioranescu D., Damlamian A. and Griso G., Periodic unfolding and homogenisation. C. R. Math. Acad. Sci. Paris 335(1) 2002, 99104.
9.Cooper S., Two-scale homogenisation of partially degenerating PDEs with applications to photonic crystals and elasticity. PhD Thesis, University of Bath, 2012.
10.Dautray R. and Lions J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3 (Spectral Theory and Applications), Springer (Berlin, 1990).
11.Evans L., Partial Differential Equations (Graduate Studies in Mathematics 19), American Mathematical Society (Providence, RI, 2010).
12.Figotin A. and Kuchment P., Spectral properties of classical waves in high-contrast periodic media. SIAM J. Appl. Math. 58 1998, 683702.
13.Hempel R. and Lienau K., Spectral properties of periodic media in the large coupling limit. Comm. Partial Differential Equations 25 2000, 14451470.
14.Jikov V. V., Kozlov S. M. and Oleinik O. A., Homogenization of Differential Operators and Integral Functionals, Springer (1994).
15.Joannopoulos J. D., Meade R. D. and Winn J. N., Photonic Crystals: Molding the Flow of Light, Princeton University Press (Princeton, NJ, 1995).
16.Kamotski I. V. and Smyshlyaev V. P., 2006, Localised modes due to defects in high contrast periodic media via homogenisation. BICS preprint 3/06, 2006, available online at: http://www.bath.ac.uk/mathsci/bics/preprints/bics063.pdf.
17.Kamotski I. V. and Smyshlyaev V. P., Two-scale homogenization for a class of partially degenerating PDE systems. Preprint, 2013, arXiv:1309.4579v1.
18.Kozlov V. A., Maz’ya V. G. and Movchan A. B., Asymptotic Analysis of Fields in Multi-Structures, Oxford University Press (Oxford, 1999).
19.Kuchment P., Floquet Theory for Partial Differential Equations (Operator Theory: Advances and Applications 60), Birkhäuser (Basel, 1993).
20.Maz’ya V. G., Sobolev Spaces, Springer (Berlin–Tokyo, 1985).
21.Maz’ya V. G., Nazarov S. A. and Plamenevskii B. A., Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. II (Operator Theory: Advances and Applications 112), Birkhäuser (Basel, 2000).
22.Nguetseng G., A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20(3) 1989, 608623.
23.Parnovski L., Bethe–Sommerfeld conjecture. Ann. Henri Poincaré 9(3) 2008, 457508.
24.Pastukhova S. E., Asymptotic analysis of elasticity problems on thin periodic structures. Netw. Heterog. Media 4(3) 2009, 577604.
25.Russell P., Photonic crystal fibers. Science 299 2003, 358362.
26.Sandrakov G. V., Averaging of non-stationary equations with contrast coefficients. Dokl. Akad. Nauk 355(5) 1997, 605608.
27.Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press (Princeton, NJ, 1970).
28.Zhikov V. V., On an extension of the method of two-scale convergence and its applications. Sb. Math. 191(7) 2000, 9731014.
29.Zhikov V. V., On gaps in the spectrum of some divergent elliptic operators with periodic coefficients. St. Petersburg Math. J. 16(5) 2004, 719773.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 86 *
Loading metrics...

Abstract views

Total abstract views: 181 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st November 2017. This data will be updated every 24 hours.