Skip to main content
SpringerLink
Log in

Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

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. V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems in Domains with a Fine-Grained Boundary [in Russian], Naukova Dumka, Kiev (1974).

    MATH  Google Scholar 

  2. E. Sanchez-Palencia, Homogenization Techniques for Composite Media, Springer-Verlag, Berlin and New York (1987).

    MATH  Google Scholar 

  3. G. A. Iosif'yan, O. A. Oleînik, and A. S. Shamaev, Mathematical Problems in the Theory of Strongly Inhomogeneous Elastic Media [in Russian], Moscow Univ., Moscow (1990).

    MATH  Google Scholar 

  4. V. V. Zhikov and S. M. Kozlov, and O. A. Oleînik, Averaging of Differential Operators [in Russian], Nauka, Moscow (1993).

    Google Scholar 

  5. D. Cioranescu and J. Saint Jean Paulin, “Truss structures: Fourier conditions and eigenvalue problems,” in: Boundary Control and Boundary Variation, Springer-Verlag, Berlin and New York, 1992, pp. 125–141 (Lecture Notes in Control and Inform. Sci.,178).

    Chapter  Google Scholar 

  6. D. Cioranescu and P. Donato, “On a Robin problem in perforated domains,” in: Homogenization and Applications to Material Sciences, Gakkōtosho, Tokyo, 1997, pp. 123–136. (GAKUTO Internat. Ser. Math. Sci. Appl.,9.

    Google Scholar 

  7. D. Cioranescu and P. Donato, “Homogénésation du problème de Neumann non homogène dans des ouverts perforés,” Asymptotic Anal.,1, No. 2, 115–138 (1988).

    MATH  Google Scholar 

  8. O. A. Oleinik and T. A. Shaposhnikova, “On the homogenization of the Poisson equation in partially perforated domain with the arbitrary density of cavities and mixed conditions on their boundary,” Matematica e Applicazioni, Rendiconti Lincei Ser. IX,8, No. 3, 129–146 (1997).

    Google Scholar 

  9. E. Sanchez-Palencia and P. Suquet, “Friction and homogenization of a boundary,” in: Free Boundary Problems. Theory and Applications, Pitman, London, 1983, pp. 561–571.

    Google Scholar 

  10. G. Bouchitte, A. Lidouh, and P. Suquet, “Homogénéisation de frontière pour la modélisation du contact entre un corps déformable non linéaire et un corps rigide,” C. R. Acad. Sci. Paris Ser. I Math.,313, No. 13, 967–972 (1991).

    MATH  Google Scholar 

  11. G. Bouchitte, A. Lidouh, J. C. Michel, and P. Suquet, Might Boundary Homogenization Help to Understand Friction? International Centre for Theoretical Physics, Trieste (1993). SMR. 719/9.

    Google Scholar 

  12. E. Sanchez-Palencia Inhomogeneous Media and Vibration Theory [Russian translation], Mir, Moscow (1984).

    Google Scholar 

  13. A. G. Belyaev, “Averaging the third boundary value problem for the Poisson equation in a domain with a rapidly oscillating boundary,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 6, 63–66 (1988).

    Google Scholar 

  14. A. G. Belyaev, On Singular Perturbations of Boundary Problems [in Russian], Dis. Kand. Fiz.-Mat. Nauk, Moscow Univ., Moscow (1990).

  15. A. B. Movchan and S. A. Nazarov, “The influence of small surface irregularities on the stress state of a body and the energy balance for a growing crack,” Prikl. Mat. Mekh.,55, No. 5, 819–828 (1991).

    Google Scholar 

  16. A. G. Belyaev, A. G. Mikheev, and A. S. Shamaev, “Plane wave diffraction by a rapoidly oscillating surface,” Zh. Vychisl. Mat. i Mat. Fiz.,32, No. 8, 1258–1272 (1992).

    MATH  Google Scholar 

  17. G. A. Chechkin, A. Friedman, and A. L. Piatnitski, The Boundary Value Problem in Domains with Very Rapidly Oscillating Boundary, INRIA Rapport de Recherche No. 3062, Unité de Recherche–Institut National de Recherche en Informatique et en Automatique, Sophia Antipolis (1996).

  18. F. Murat and L. Tartar, Calcul des Variations et Homogénéisation, Université Pierre et Marie Curie, Centre National de la Recherche Scientifique, Laboratoire d'Analyse Numérique, R 84012. Paris (1984).

  19. G. Allaire, “Homogenization and two-scale convergence,” SIAM J. Math. Anal.,23, 1482–1518 (1992).

    Article  MATH  Google Scholar 

  20. N. S. Bakhvalov, “Averaged characteristics of bodies with periodic structure,” Dokl. Akad. Nauk SSSR,218, No. 5, 1046–1048 (1974).

    Google Scholar 

  21. N. S. Bakhvalov, “Averaging of partial differential equations with rapidly oscillating coefficients,” Dokl. Akad. Nauk SSSR,221, No. 3, 516–519 (1975).

    Google Scholar 

  22. J.-L. Lions and E. Magenes, Inhomogeneous Boundary Value Problems and Their Applications [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  23. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  24. M. Briane, Homogénéisation de Matériaux Fibres et Multi-Couches, PhD Thesis, University Paris 6, Paris (1990).

    Google Scholar 

  25. J. M. Ball and F. Murat, Remarks on Rank-One Convexity and Quasiconvexity, Université Pierre et Marie Curie, Centre National de la Recherche Scientifique, Laboratoire d'Analyse Numérique. Paris (1990). Report 90043.

    Google Scholar 

  26. E. M. Landis and G. P. Panasenko, “A theorem on the asymptotic behavior of the solutions of elliptic equations with coefficients that are periodic in all variables, except one,” Dokl. Akad. Nauk SSSR,235, No. 6, 1253–1255 (1977).

    Google Scholar 

  27. O. A. Oleînik and G. A. Iosif'yan, “On the behavior at infinity of solutions to second-order elliptic equations in domains with noncompact boundary,” Mat. Sb.,112, No. 4, 588–610 (1980).

    Google Scholar 

Download references

Authors
  1. A. G. Belyaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. L. Pyatnitskiî
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. G. A. Chechkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

To the unfaded memory of Sergeî L'vovich sobolev, an outstanding contemporary mathematician.

The second and third authors were financially supported by the Russian Foundation for Basic Research (Grant 98-01-00062).

Aizu (Japan). Moscow. Translated fromSibirskiî Matematicheskiî Zhurnal. Vol. 39, No. 4, pp. 730–754. July–August, 1998.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Belyaev, A.G., Pyatnitskiî, A.L. & Chechkin, G.A. Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary. Sib Math J 39, 621–644 (1998). https://doi.org/10.1007/BF02673049

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02673049

Keywords

Search

Navigation