Skip to main content
SpringerLink
Log in

Asymptotic expansions of eigenvalues and eigenfunctions for elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We consider spectral boundary value problems of Steklov, Neumann, and Dirichlet types for second-order elliptic operators with ε-periodic coefficients in a perforated cube; the coefficients of the differential equations are assumed to satisfy some symmetry conditions. Complete asymptotic expansions with respect to the small parameter ε are constructed for eigenvalues and eigenfunctions of the said problems. Bibliography: 24 titles.

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

Similar content being viewed by others

Literature Cited

  1. N.S. Bakhvalov, and G.P. Panasenko,Homogenization of Processes in Periodic Media, Nauka, Moscow, 1984. (in Russian).

    Google Scholar 

  2. V.V. Zhikov, S.N. Kozlov, and O.A. Oleinik,Homogenization of Differential Operators, Nauka, Moscow, 1984. (in Russian).

    Google Scholar 

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

    Google Scholar 

  4. E. Sanchez-Palencia,Non-Homogeneous Media and Vibration Problems, Lecture Notes in Physics,127, Springer-Verlag, Berlin, 1980.

    Google Scholar 

  5. O.A. Oleinik, G.P. Panasenko, and G.A. Yosifian,Asymptotic expansion of solutions for the system of linear elasticity in perforated domains, Mat. Sb.120, (1983), No. 1, 22–41.

    MathSciNet  Google Scholar 

  6. S.A. Nazarov and M.V. Paukshto,Discrete Models and Homogenization in Problems of Elasticity, Leningrad Univ. Press, Leningrad, 1984. (in Russian).

    Google Scholar 

  7. M.I. Vishik and L.A. Lyusternik,Regular degeneration and boundary layers for linear differential equations with a small parameter, Usp. Mat. Nauk52 (1957), no. 5, 3–122.

    MathSciNet  Google Scholar 

  8. O.A. Oleinik, A.S. Shamaev, and G.A. Yosifian,Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992.

    Google Scholar 

  9. G.A. Yosifian, O.A. Oleinik, and A.S. Shamaev,On eigenvalues of boundary value problems for the system of elasticity with rapidly oscillating coefficients in a perforated domain, Mat. Sb.132 (1987), No. 4, 517–530.

    Google Scholar 

  10. G.A. Yosifian, O.A. Oleinik, and A.S. Shamaev,Asymptotic expansions for eigenvalues and eigenfunctions of the Sturm-Liouville problem with rapidly oscillating coefficients, Vestn. Mosk. Univ.6 (1985), 22–35.

    Google Scholar 

  11. G.A. Yosifian, O.A. Oleinik, and A.S. Shamaev,On the convergence of the energy, stress tensors, and frequencies of free vibrations in homogenization problems arising in elasticity, Dokl. Akad. Nauk SSSR274 (1984), no. 6, 1329–1333.

    MathSciNet  Google Scholar 

  12. A.S. Shamaev,Spectral problems in the theory of homogenization and G-convergence, Dokl. Akad. Nauk SSSR252 (1981), no. 2, 294–299.

    MathSciNet  Google Scholar 

  13. A.G. Belyaev,Homogenization of the third boundary value problem for the Poisson equation in a domain with rapidly oscillating boundary, Vestn. Mosk. Univ. Ser. Mat. Mekh. (1988), no. 6, 63–66.

    MATH  MathSciNet  Google Scholar 

  14. O.A. Oleinik, and G.A. Yosifian,On the homogenization of the system of elasticity with rapidly oscillating coefficients in perforated domains, N.E. Kochin and Advances in Mechanics, Nauka, Moscow, 1982. (in Russian).

    Google Scholar 

  15. T.A. Melnik,On asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary value problems with rapidly oscillating coefficients, Usp. Mat. Nauk42 (1987), no. 5, 167.

    MathSciNet  Google Scholar 

  16. T.A. Melnik,Asymptotic expansions for eigenvalues and eigenfunctions of the Dirichlet problem in a perforated cube, Registered at VINITI, 7.12.87, No. 8567-B87.

  17. T.A. Melnik,Asymptotic expansions for eigenvalues and eigenfunctions of a strongly non-homogeneous plate with a sliding edge, Proceedings of the All-Union Conference on Nonlinear Problems of Differential Equations and Mathematical Physics, September 12–15, 1989, pp. 280–281.

  18. O.A. Oleinik, A.S. Shamaev, and G.A. Yosifian,Homogenization of eigenvalues and eigenfunctions of boundary value problems in perforated domains for elliptic equations with non-uniformly oscillating coefficients, Current Topics in Partial Differential Equations, Kinokuniya, Tokyo, 1986, pp. 187–216.

  19. A. Bensoussan, J.-L. Lions, and G. Papanicolaou,Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.

    Google Scholar 

  20. L. Tartar,Course Peccot, College de France, Paris, 1977.

    Google Scholar 

  21. M. Vanninathan,Homogénéisation des valeurs propres dans les milieux perforées, C.R. Acad. Sci. ParisA 287 (1978), no. 6, 403–406.

    MATH  MathSciNet  Google Scholar 

  22. M. Vanninathan,Homogénéisation des problemes de valeurs propers dans les milieux perforées. Probleme de Dirichlet, C.R. Acad. Sci. ParisA287 (1978), no. 12, 823–825.

    MATH  MathSciNet  Google Scholar 

  23. S. Kesavan,Homogenization of elliptic eigenvalue problems, Part 1, Appl. Math. Optim.5 (1979), no. 2, 153–167; Part 25 (1979), no. 3, 197–216.

    MATH  MathSciNet  Google Scholar 

  24. D. Cioranescu, and S.-J. Paulin,Homogenization in open sets with holes, J. App. Math. Anal. Appl.71 (1979), no. 2, 590–607.

    MathSciNet  Google Scholar 

Download references

Authors
  1. T. A. Melnik
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 51–88, 1994.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Melnik, T.A. Asymptotic expansions of eigenvalues and eigenfunctions for elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube. J Math Sci 75, 1646–1671 (1995). https://doi.org/10.1007/BF02368668

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02368668

Keywords

Search

Navigation