Skip to main content
SpringerLink
Log in

Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

  • Brief Communications
  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

Let O ⊂ Rd be a bounded domain of class C 1,1. Let 0 < ε - 1. In L 2(O;Cn) we consider a positive definite strongly elliptic second-order operator B D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B D,ε − ζQ 0(·/ε))−1 as ε → 0. Here the matrix-valued function Q 0 is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L 2(O;Cn)-operator norm and in the norm of operators acting from L 2(O;Cn) to the Sobolev space H 1(O;Cn) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q 0(x/ε)∂ t v ε (x, t) = −(B D,ε v ε )(x, t).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, Studies in Math. and Appl., vol. 5, North-Holland, Amsterdam–New York, 1978.

    MATH  Google Scholar 

  2. N. S. Bakhvalov, 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.

    MATH  Google Scholar 

  3. E. Sanchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Phys., vol. 127, Springer-Verlag, Berlin–New York, 1980.

    Google Scholar 

  4. V. V. Zhikov, S. M. Kozlov, O. A. Olejnik, Homogenization of differential operators, Springer-Verlag, Berlin, 1994.

    MATH  Google Scholar 

  5. M. Sh. Birman, T. A. Suslina, Algebra i Analiz, 15:5 (2003), 1–108; St. Petersburg Math. J., 15:5 (2004), 639–714.

    MathSciNet  Google Scholar 

  6. M. Sh. Birman, T. A. Suslina, Algebra i Analiz, 18:6 (2006), 1–130; St. Petersburg Math. J., 18:6 (2007), 857–955.

    MathSciNet  Google Scholar 

  7. V. V. Zhikov, S. E. Pastukhova, Russian J. Math. Phys., 12:4 (2005), 515–524.

    MathSciNet  Google Scholar 

  8. V. V. Zhikov, S. E. Pastukhova, Uspekhi Mat. Nauk, 71:3 (2016), 27–122; Russian Math. Surveys, 71:3 (2016), 417–511.

    Article  MathSciNet  Google Scholar 

  9. D. I. Borisov, Algebra i Analiz, 20:2 (2008), 19–42; St. Petersburg Math. J., 20:2 (2009), 175–191.

    MathSciNet  Google Scholar 

  10. T. A. Suslina, Algebra i Analiz, 22:1 (2010), 108–222; St. Petersburg Math. J., 22:1 (2011), 81–162.

    MathSciNet  Google Scholar 

  11. T. A. Suslina, Algebra i Analiz, 27:4 (2015), 87–166; St. Petersburg Math. J., 27:4 (2016), 651–708.

    MathSciNet  Google Scholar 

  12. Yu. M. Meshkova, T. A. Suslina, Appl. Anal., 95:7 (2016), 1413–1448.

    Article  MathSciNet  Google Scholar 

  13. G. Griso, Asymptot. Anal., 40:3/4 (2004), 269–286.

    MathSciNet  Google Scholar 

  14. G. Griso, Anal. Appl., 4:1 (2006), 61–79.

    Article  MathSciNet  Google Scholar 

  15. C. E. Kenig, F. Lin, Z. Shen, Arch. Rational Mech. Anal., 203:3 (2012), 1009–1036.

    Article  MathSciNet  Google Scholar 

  16. M. A. Pakhnin, T. A. Suslina, Algebra i Analiz, 24:6 (2012), 139–177; St. Petersburg Math. J., 24:6 (2013), 949–976.

    MathSciNet  Google Scholar 

  17. T. A. Suslina, Mathematika, 59:2 (2013), 463–476.

    Article  MathSciNet  Google Scholar 

  18. T. A. Suslina, SIAM J. Math. Anal., 45:6 (2013), 3453-3493.

    Google Scholar 

  19. T. A. Suslina, Funkts. Anal. Prilozhen., 38:4 (2004), 86–90; Functional Anal. Appl., 38:4 (2004), 309–312.

    Article  Google Scholar 

  20. T. A. Suslina, Math. Model. Nat. Phenom., 5:4 (2010), 390-447.

    Google Scholar 

  21. V. V. Zhikov and S. E. Pastukhova, Russian J. Math. Phys., 13:2 (2006), 224–237.

    Article  MathSciNet  Google Scholar 

  22. Yu. M. Meshkova, Algebra i Analiz, 25:6 (2013), 125–177; St. Petersburg Math. J., 25:6 (2014), 981–1019.

    MathSciNet  Google Scholar 

  23. Yu. M. Meshkova, T. A. Suslina, Appl. Anal., 95:8 (2016), 1736–1775.

    Article  MathSciNet  Google Scholar 

  24. J. Geng, Zh. Shen, J. Funct. Anal., 272:5 (2017), 2092–2113.

    Article  MathSciNet  Google Scholar 

  25. Q. Xu, SIAM J. Math. Anal., 48:6 (2016), 3742–3788.

    Article  MathSciNet  Google Scholar 

  26. W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge Univ. Press, Cambridge, 2000.

    MATH  Google Scholar 

  27. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, 1970.

    MATH  Google Scholar 

  28. Yu. M. Meshkova, T. A. Suslina, http://arxiv.org/abs/1702.00550.

  29. Yu. M. Meshkova, T. A. Suslina, PDMI preprint 6/2017; http://www.pdmi.ras.ru/preprint/2017/rus-2017.html.

Download references

Author information

Authors and Affiliations

  1. Chebyshev Laboratory, St. Petersburg State University, St. Petersburg, Russia

    Yu. M. Meshkova

  2. Department of Physics, St. Petersburg State University, St. Petersburg, Russia

    T. A. Suslina

Authors
  1. Yu. M. Meshkova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. A. Suslina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu. M. Meshkova.

Additional information

To the blessed memory of Mikhail Semenovich Agranovich

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 51, No. 3, pp. 87–93, 2017

Original Russian Text Copyright © by Yu. M. Meshkova and T. A. Suslina

Supported by RFBR (project no. 16-01-00087). The first author is supported by “Native Towns,” a social investment program of PJSC “Gazprom Neft,” by the “Dynasty” foundation, and by the Rokhlin grant.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meshkova, Y.M., Suslina, T.A. Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients. Funct Anal Its Appl 51, 230–235 (2017). https://doi.org/10.1007/s10688-017-0187-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10688-017-0187-y

Key words

Search

Navigation