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Mathematical Notes

, Volume 102, Issue 5–6, pp 645–663 | Cite as

Homogenization of a nonstationary model equation of electrodynamics

  • M. A. DorodnyiEmail author
  • T. A. Suslina
Article
  • 30 Downloads

Abstract

In L 2(ℝ3;ℂ3), we consider a self-adjoint operator ℒ ε , ε > 0, generated by the differential expression curl η(x/ε)−1 curl−∇ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ ε 1/2 ) and ℒ ε −1/2 sin(τ ε 1/2 ) for τ ∈ ℝ and small ε. It is shown that these operators converge to cos(τ(ℒ0)1/2) and (ℒ0)−1/2 sin(τ(ℒ0)1/2), respectively, in the norm of the operators acting from the Sobolev space H s (with a suitable s) to ℒ2. Here ℒ0 is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation τ 2 v ε = −ℒ ε v ε , div v ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).

Keywords

periodic differential operator homogenization operator error estimate nonstationary Maxwell system 

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References

  1. 1.
    A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, in Stud. Math. Appl. (North-Holland Publ., Amsterdam, 1978), Vol. 5.Google Scholar
  2. 2.
    N. S. Bakhvalov and G. P. Panasenko, Homogenization of Processes in Periodic Media. Mathematical Problems of the Mechanics of Composite Materials (Nauka, Moscow, 1984; Kluwer Acad. Publ. Group, Dordrecht, 1989).zbMATHGoogle Scholar
  3. 3.
    V. V. Zhikov, S. M. Kozlov, and O. A. Olejnik (Oleinik), Homogenization of Differential Operators (Fizmatlit, Moscow, 1993; Springer-Verlag, Berlin, 1994).Google Scholar
  4. 4.
    M. Sh. Birman and T. A. Suslina, “Second-order periodic differential operators. Threshold properties and homogenization,” Algebra Anal. 15 (5), 1–108 (2003) [St. PetersburgMath. J. 15 (5), 639–714 (2004)].zbMATHGoogle Scholar
  5. 5.
    M. Sh. Birman and T. A. Suslina, “Homogenization with corrector term for periodic elliptic differential operators,” Algebra Anal. 17 (6), 1–104 (2005) [St. PetersburgMath. J. 17 (6), 897–973 (2006)].zbMATHGoogle Scholar
  6. 6.
    M. Sh. Birman and T. A. Suslina, “Homogenization with corrector term for periodic differential operators. Approximation of solutions in the Sobolev class H1(Rd),” Algebra Anal. 18 (6), 1–130 (2006) [St. Petersburg Math. J. 18 (6), 857–955 (2007)].Google Scholar
  7. 7.
    M. Sh. Birman and T. A. Suslina, “Operator error estimates in the homogenization problem for nonstationary periodic equations,” Algebra Anal. 20 (6), 30–107 (2008) [St. PetersburgMath. J. 20 (6), 873–928 (2009)].Google Scholar
  8. 8.
    T. A. Suslina, “On homogenization of periodic parabolic systems,” Funktsional. Anal. Prilozhen. 38 (4), 86–90 (2004) [Functional Anal. Appl. 38 (4), 309–312 (2004)].MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    V. V. Zhikov, “On operator estimates in homogenization theory,” Dokl. Akad. Nauk 403 (3), 305–308 (2005) [Dokl.Math. 72 (1), 534–538 (2005)].MathSciNetzbMATHGoogle Scholar
  10. 10.
    V. V. Zhikov and S. E. Pastukhova, “On operator estimates for some problems in homogenization theory,” Russ. J. Math. Phys. 12 (4), 515–524 (2005).MathSciNetzbMATHGoogle Scholar
  11. 11.
    V. V. Zhikov and S. E. Pastukhova, “Estimates of homogenization for a parabolic equation with periodic coefficients,” Russ. J. Math. Phys. 13 (2), 224–237 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    V. V. Zhikov and S. E. Pastukhova, “Operator estimates in homogenization theory,” UspekhiMat. Nauk 71 (3 (429)), 27–122 (2016) [RussianMath. Surveys 71 (3), 417–511 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    T. A. Suslina, “Homogenization of Schrödinger-type equations,” Funktsional. Anal. Prilozhen. 50 (3), 90–96 (2016) [Functional Anal. Appl. 50 (3), 241–246 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    T. A. Suslina, “Spectral approach to homogenization of nonstationary Schrödinger-type equations,” J.Math. Anal. Appl. 446 (2), 1466–1523 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yu. M. Meshkova, On Operator Error Estimates for Homogenization of Hyperbolic Systems with Periodic Coefficients, arXiv: https://arxiv.org/abs/1705.02531v3 (2017).Google Scholar
  16. 16.
    M. A. Dorodnyi and T. A. Suslina, “Homogenization of hyperbolic equations,” Funktsional. Anal. Prilozhen. 50 (4), 91–96 (2016) [Functional Anal. Appl. 50 (4), 319–324 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    M. A. Dorodnyi and T. A. Suslina, Spectral Approach to Homogenization of Hyperbolic Equations with Periodic Coefficients, arXiv: https://arxiv.org/abs/1708.00859.Google Scholar
  18. 18.
    T. A. Suslina, “Homogenization of a stationary periodic Maxwell system,” Algebra Anal. 16 (5), 162–244 (2004) [St. PetersburgMath. J. 16 (5), 863–922 (2004)].Google Scholar
  19. 19.
    T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Heidelberg, 1966; Mir, Moscow, 1972).CrossRefzbMATHGoogle Scholar
  20. 20.
    V. V. Zhikov, “Estimates for the averaged matrix and the averaged tensor,” UspekhiMat. Nauk 46 (3 (279)), 49–109 (1991) [Russian Math. Surveys 46 (3), 65–136 (1991)].MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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