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Ukrainian Mathematical Journal

, Volume 61, Issue 1, pp 121–139 | Cite as

On Γ-convergence of integral functionals defined on various weighted Sobolev spaces

  • O. A. Rudakova
Article

We consider weighted Sobolev spaces correlated with a sequence of n-dimensional domains. We prove a theorem on the choice of a subsequence Γ-convergent to an integral functional defined on a “limit” weighted Sobolev space from a sequence of integral functionals defined on the spaces indicated.

Keywords

Banach Space Sobolev Space Variational Problem Integral Functional Weighted Sobolev Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    E. de Giorgi and T. Franzoni, “Su un tipo di convergenza variazionale,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis., Mat. Natur., 58, No. 6, 842–850 (1975).zbMATHMathSciNetGoogle Scholar
  2. 2.
    C. Sbordone, “Su alcune applicazioni di un tipo di convergenza variazionale,” Ann. Scuola Norm. Super. Pisa Cl. Sci., 2, 617–638 (1975).zbMATHMathSciNetGoogle Scholar
  3. 3.
    G. dal Maso, An Introduction to Γ-Convergence, Birkhäuser, Boston (1993).Google Scholar
  4. 4.
    A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Clarendon Press, New York (1998).zbMATHGoogle Scholar
  5. 5.
    V. V. Zhikov, “Problems of convergence, duality, and averaging for one class of functionals of variational calculus,” Dokl. Akad. Nauk SSSR, 267, No. 3, 524–528 (1982).MathSciNetGoogle Scholar
  6. 6.
    V. V. Zhikov, “Problems of convergence, duality, and averaging for a functional of variational calculus,” Izv. Akad. Nauk SSSR, Ser. Mat., 47, No. 5, 961–998 (1983).zbMATHMathSciNetGoogle Scholar
  7. 7.
    V. V. Zhikov, “Averaging of functionals of variational calculus and theory of elasticity,” Izv. Akad. Nauk SSSR, Ser. Mat., 50, No. 4, 675–710 (1986).MathSciNetGoogle Scholar
  8. 8.
    V. V. Zhikov, “On limit transition in nonlinear variational problems,” Mat. Sb., 183, No. 8, 47–84 (1992).zbMATHGoogle Scholar
  9. 9.
    A. A. Kovalevskii, “Averaging of variable variational problems,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 8, 6–9 (1988).Google Scholar
  10. 10.
    A. A. Kovalevskii, “On correlation of subsets of Sobolev spaces and Γ-convergence of functionals with variable domain of definition,” Nelin. Gran. Zad., Issue 1, 48–54 (1989).Google Scholar
  11. 11.
    A. A. Kovalevskii, “On some problems related to the problem of averaging of variational problems for functionals with variable domain of definition,” in: Contemporary Analysis and Its Applications [in Russian], Naukova Dumka, Kiev (1989), pp. 62–70.Google Scholar
  12. 12.
    A. A. Kovalevskii, “Conditions for Γ-convergence and averaging of integral functionals with different domains of definition,” Dokl. Akad. Nauk Ukr. SSR, No. 4, 5–8 (1991).Google Scholar
  13. 13.
    A. A. Kovalevskii, “On necessary and sufficient conditions for the Γ-convergence of integral functionals with different domains of definition,” Nelin. Gran. Zad., Issue 4, 29–39 (1992).Google Scholar
  14. 14.
    A. A. Kovalevskii, “On Γ-convergence of integral functionals defined on weakly correlated Sobolev spaces,” Ukr. Mat. Zh., 48, No. 5, 614–628 (1996).CrossRefMathSciNetGoogle Scholar
  15. 15.
    L. S. Pankratov, “Γ-convergence of nonlinear functionals in thin reticulated structures,” C. R. Acad. Sci. Paris, Ser. I, 335, No. 3, 315–320 (2002).zbMATHMathSciNetGoogle Scholar
  16. 16.
    B. Amaziane, M. Goncharenko, and L. Pankratov, “ΓD-convergence for a class of quasilinear elliptic equations in thin structures,” Math. Meth. Appl. Sci., 28, No. 15, 1847–1865 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    A. Kovalevsky and F. Nicolosi, “On the convergence of solutions of degenerate nonlinear elliptic high order equations,” Nonlin. Anal., Theory, Methods, Appl., 49, 335–360 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    E. Ya. Khruslov, “Asymptotic behavior of solutions of the second boundary-value problem in the case of refinement of the boundary of the domain,” Mat. Sb., 106, No. 4, 604–621 (1978).MathSciNetGoogle Scholar
  19. 19.
    E. Ya. Khruslov, “On the convergence of solutions of the second boundary-value problem in weakly correlated domains,” in: Theory of Operators in Functional Spaces and Its Applications [in Russian], Naukova Dumka, Kiev (1981), pp. 129–173.Google Scholar
  20. 20.
    L. V. Berlyand and I. Yu. Chudinovich, “Averaging of boundary-value problems for higher-order differential operators in domains with voids,” Dokl. Akad. Nauk SSSR, 272, No. 4, 777–780 (1983).MathSciNetGoogle Scholar
  21. 21.
    L. S. Pankratov, On Convergence of Solutions of Variational Problems in Weakly Correlated Domains [in Russian], Preprint No. 53.88, Institute for Low Temperature Physics and Engineering, Academy of Sciences of Ukr. SSR, Kharkov (1988).Google Scholar
  22. 22.
    A. A. Kovalevskii, “G-convergence and averaging of nonlinear elliptic operators of divergence type with variable domain of definition,” Izv. Ros. Akad. Nauk, Ser. Mat., 58, No. 3, 3–35 (1994).MathSciNetGoogle Scholar
  23. 23.
    V. A. Marchenko and E. Ya. Khruslov, Averaged Models of Microinhomogeneous Media [in Russian], Naukova Dumka, Kiev (2005).Google Scholar
  24. 24.
    A. A. Kovalevskii and O. A. Rudakova, “On the strong correlation of weighted Sobolev spaces and compactness of sequences of their elements,” Tr. Inst. Prikl. Mat. Mekh. Nats. Akad. Nauk Ukr., 12, 85–99 (2006).MathSciNetGoogle Scholar
  25. 25.
    R. de Arcangelis and P. Donato, “Homogenization in weighted Sobolev spaces,” Ric. Mat., 34, 289–308 (1985).zbMATHGoogle Scholar
  26. 26.
    R. de Arcangelis and F. S. Cassano, “On the convergence of solutions of degenerate elliptic equations in divergence form,” Ann. Mat. Pura Appl., 167, 1–23 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    I. V. Skrypnik and D. V. Larin, “Principle of additivity in averaging of degenerate nonlinear Dirichlet problems,” Ukr. Mat. Zh., 50, No. 1, 118–135 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    D. V. Larin, “On convergence of solutions of a degenerate quasilinear Dirichlet problem in the case of refinement of the boundary of the domain,” Dopov. Nats. Akad. Nauk Ukr., No. 8, 37–41 (1998).Google Scholar
  29. 29.
    D. V. Larin, “Homogenization of degenerate nonlinear Dirichlet problems in perforated domains of general structure,” Nelin. Gran. Zad., Issue 10, 117–122 (2000).Google Scholar
  30. 30.
    A. A. Kovalevskii and O. A. Rudakova, “On Γ-compactness of integral functionals with degenerate Lagrangians,” Nelin. Gran. Zad., Issue 15, 149–153 (2005).Google Scholar
  31. 31.
    S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics [in Russian], Nauka, Moscow (1988).Google Scholar
  32. 32.
    M. K. V. Murthy and G. Stampacchia, “Boundary-value problem for some degenerate elliptic operators,” Ann. Mat. Pura Appl., 80, 1–122 (1969).MathSciNetGoogle Scholar
  33. 33.
    F. Guglielmino and F. Nicolosi, “Sulle W-soluzioni dei problemi al contorno per operatori ellittici degeneri,” Ric. Mat., 36, 59–72 (1987).MathSciNetGoogle Scholar
  34. 34.
    G. R. Cirmi and M. M. Porzio, “L -solutions for some nonlinear degenerate elliptic and parabolic equations,” Ann. Mat. Pura Appl., 169, 67–86 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    A. Kovalevsky and F. Nicolosi, “Boundedness of solutions of variational inequalities with nonlinear degenerate elliptic operators of high order,” Appl. Anal., 65, 225–249 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford (1993).zbMATHGoogle Scholar
  37. 37.
    O. A. Rudakova, “On the coercivity of the Lagrangian of the Γ-limit functional of a sequence of integral functionals defined on different weighted Sobolev spaces,” Tr. Inst. Prikl. Mat. Mekh. Nats. Akad. Nauk Ukr., 15, 171–180 (2007).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • O. A. Rudakova
    • 1
  1. 1.Institute of Applied Mathematics and MechanicsUkrainian National Academy of SciencesDonetskUkraine

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