Functional Analysis and Its Applications

, Volume 41, Issue 2, pp 81–98

Homogenization of the stationary periodic Maxwell system in the case of constant permeability

  • M. Sh. Birman
  • T. A. Suslina
Article

Abstract

The homogenization problem in the small period limit for the stationary periodic Maxwell system in ℝ3 is considered. It is assumed that the permittivity ηε(x)=η(εx), ε > 0, is a rapidly oscillating positive matrix function and the permeability µ0 is a constant positive matrix. For all four physical fields (the electric and magnetic field intensities, the electric displacement field, and the magnetic flux density), we obtain uniform approximations in the L2(ℝ3)-norm with order-sharp remainder estimates.

Key words

periodic Maxwell operator homogenization effective medium corrector 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging Processes in Periodic Media, Mathematical Problems in the Mechanics of Composite Materials, Math. Appl. (Soviet Ser.), vol. 36, Kluwer Acad. Publ. Group, Dordrecht, 1989.Google Scholar
  2. [2]
    A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Co., Amsterdam-New York, 1978.MATHGoogle Scholar
  3. [3]
    M. Sh. Birman and T. A. Suslina, “Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics,” in: Oper. Theory Adv. Appl., vol. 129, Birkhauser, Basel, 2001, 71–107.Google Scholar
  4. [4]
    M. Sh. Birman and T. A. Suslina, “Second-order periodic differential operators. Threshold properties and homogenization,” Algebra i Analiz, 15:5 (2003), 1–108; English transl.: St. Petersburg Math. J., 15:5 (2004), 639–714.Google Scholar
  5. [5]
    M. Sh. Birman and T. A. Suslina, “Threshold approximations with the corrector term for the resolvent of a factorized selfadjoint operator family,” Algebra i Analiz, 17:5 (2005), 69–90; English transl.: St. Petersburg Math. J., 17:5 (2006), 745–762.Google Scholar
  6. [6]
    M. Sh. Birman and T. A. Suslina, “Homogenization with the corrector term for periodic elliptic differential operators,” Algebra i Analiz, 17:6 (2005), 1–104; English transl.: St. Petersburg Math. J., 17:6 (2006), 897–973.Google Scholar
  7. [7]
    M. Sh. Birman and T. A. Suslina, “Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class H 1(ℝd),” Algebra i Analiz, 18:6 (2006), 1–130; English transl.: St. Petersburg Math. J., 18:6 (2007).Google Scholar
  8. [8]
    V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators, Springer-Verlag, Berlin, 1994.MATHGoogle Scholar
  9. [9]
    E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Phys., vol. 127, Springer-Verlag, Berlin-New York, 1980.Google Scholar
  10. [10]
    T. A. Suslina, “On homogenization of periodic Maxwell system,” Funkts. Anal. Prilozhen., 38:3 (2004), 90–94; English transl.: Funct. Anal. Appl., 38:3 (2004), 234–237.Google Scholar
  11. [11]
    T. A. Suslina, “Homogenization of a stationary periodic Maxwell system,” Algebra i Analiz, 16:5 (2004), 162–244; English transl.: St. Petersburg Math. J., 16:5 (2005), 863–922.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • M. Sh. Birman
    • 1
  • T. A. Suslina
    • 1
  1. 1.Department of PhysicsSt. Petersburg State UniversityRussia

Personalised recommendations