Skip to main content
SpringerLink
Log in

New conditions for averaging of nonlinear dirichlet problems in perforated domains

  • Published:
Ukrainian Mathematical Journal Aims and scope

Abstract

We study the problem of averaging Dirichlet problems for nonlinear elliptic second-order equations in domains with fine-grained boundary. We consider a class of equations admitting degeneration with respect to the gradients of solutions. We prove a pointwise estimate for solutions of the model nonlinear boundary-value problem and construct an averaged boundary-value problem under new structural assumptions concerning perforated domains. In particular, it is not assumed that the diameters of cavities are small as compared to the distances between them.

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

References

  1. I. V. Skrypnik, “A quasilinear Dirichlet problem in domains with fine-grained boundary,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 2, 21–25 (1982).

  2. I. V. Skrypnik,Nonlinear Elliptic Boundary Value Problems, Teubner, Leipzig 1986.

    MATH  Google Scholar 

  3. I. V. Skrypnik,Methods for Studying Nonlinear Elliptic Boundary-Value Problems [in Russian], Nauka, Moscow 1990.

    Google Scholar 

  4. I. V. Skrypnik, “Averaging of nonlinear Dirichlet problems in domains with canals,”Dokl. Akad. Nauk SSSR,315, No. 4, 793–797 (1991).

    Google Scholar 

  5. I. V. Skrypnik, “Asymptotic behavior of solutions of nonlinear elliptic problems in perforated domains,”Mat. Sb.,184, No. 10, 67–90 (1993).

    Google Scholar 

  6. I. V. Skrypnik,Homogenization of Nonlinear Dirichlet Problems in Perforated Domains of General Structure, Preprint, SISSA, Trieste (1994).

    Google Scholar 

  7. G. Dal Maso and A. Garroni,A New Approach to the Study of Limits of Dirichlet Problems in Perforated Domains, Preprint, SISSA, Trieste (1993).

    Google Scholar 

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

    Google Scholar 

  9. G. Dal Maso and A. Defranceschi, “Limits of nonlinear Dirichlet problems in varying domains,”Manuscr. Math.,61, 251–278, (1988).

    Article  MATH  Google Scholar 

  10. G. Dal Maso and F. Murat,Dirichlet Problems in Perforated Domains for Homogeneous Monotone Operators in H 01 , Preprint, SISSA, Trieste (1994).

    Google Scholar 

  11. G. Dal Maso and F. Murat,Asymptotic Behaviour and Correctors for Dirichlet Problems in Perforated Domains with Homogeneous Monotone Operators, Preprint, SISSA, Trieste (1994).

    Google Scholar 

  12. G. Dal Maso and I. V. Skrypnik,Asymptotic Behaviour of Nonlinear Dirichlet Problems in Perforated Domains, Preprint, SISSA, Trieste (1994).

    Google Scholar 

  13. A. A. Kovalevskii, “G-convergence of operators of Dirichlet problems in varying domains,”Dokl. Akad. Nauk Ukrainy, Ser. A, No. 5, 13–17 (1993).

    Google Scholar 

  14. L. S. Pankratov,On the Convergence of Solutions of Variational Problems in Weakly Connected Domains [in Russian], Preprint No. 53.88, Physicotechnical Institute of Low Temperatures, Ukrainian Academy of Sciences, Kharkov (1988).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ukrainian Academy of Sciences, Ukrainian

    I. V. Skrypnik

Authors
  1. I. V. Skrypnik
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Skrypnik, I.V. New conditions for averaging of nonlinear dirichlet problems in perforated domains. Ukr Math J 48, 753–774 (1996). https://doi.org/10.1007/BF02384225

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02384225

Keywords

Search

Navigation