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Differential Equations

, Volume 55, Issue 4, pp 523–531 | Cite as

Homogenization of a Boundary Value Problem for the n-Laplace Operator on a n-Dimensional Domain with Rapidly Alternating Boundary Condition Type: The Critical Case

  • A. V. PodolskiyEmail author
  • T. A. ShaposhnikovaEmail author
Partial Differential Equations

Abstract

We study the asymptotic behavior of the solution of a boundary value problem for the p-Laplace operator with rapidly alternating nonlinear boundary conditions posed on ε-periodically arranged subsets on the boundary of a domain Ω ⊂ ℝn. We assume that p = n, construct a homogenized problem, and prove the weak convergence as ε → 0 of the solution of the original problem to the solution of the homogenized problem in the so-called critical case, which is characterized by the fact that the homogenization changes the character of nonlinearity of the boundary condition.

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References

  1. 1.
    Zubova, M.N. and Shaposhnikova, T.A., Homogenization of boundary value problems in perforated domains with third boundary conditions and the resulting change in the character of the nonlinearity in the problem, Differ. Equations, 2011, vol. 47, no. 1, pp. 78–90.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Gomez, D., Perez, M.E., Podolskii, A.V., and Shaposhnikova, T.A., Homogenization of variational in-equalities for the p-Laplace operator in perforated media along manifolds, Appl. Math. Optim., 2017, vol. 475, pp. 1–19.Google Scholar
  3. 3.
    Kaizu, S., The Poisson equation with semilinear boundary conditions in domains with many tiny holes, J. Fac. Sci. Univ. Tokyo. Sect. IA Math., 1989, vol. 36, pp. 43–86.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Goncharenko, M., The asymptotic behaviour of the third boundary-value problem solutions in domains with fine-grained boundaries, GAKUTO Int. Ser. Math. Sci. Appl. 1995, Vol. 9: Homogenization and Applications to Material Sciences, pp. 203–213.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Diaz, J.I., Gomez-Castro, D., Shaposhnikova, T.A., and Zubova, M.N., Change of homogenized absorption term in diffusion processes with reaction on the boundary of periodically distributed asymmetric particles of critical size, Electron. J. Differ. Equ., 2017, no. 178, pp. 1–25.Google Scholar
  6. 6.
    Gomez, D., Lobo, M., Perez, E., Podol’skii, A.V., and Shaposhnikova, T.A., Unilateral problems for the p-Laplace operator in perforated media involving large parameters, ESAIM Control Optim. Calc. Var., 2018, vol. 24, no. 3, pp. 921–964.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Perez, M.E., Zubova, M.N., and Shaposhnikova, T.A., A homogenization problem in a domain perforated by tiny isoperimetric holes with nonlinear Robin type boundary conditions, Dokl. Math., 2014, vol. 90, no. 1, pp. 489–494.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Podol’skii, A.V. and Shaposhnikova, T.A., Homogenization for the p-Laplacian in an n-dimensional domain perforated by very thin cavities with a nonlinear boundary condition on their boundary in the case p = n, Dokl. Math., 2015, vol. 92, no. 1, pp. 464–470.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Diaz, J.I., Gomez-Castro, D., Podol’skii, A.V., and Shaposhnikova, T.A., Characterizing the strange term in critical size homogenization: quasilinear equations with a nonlinear boundary condition involving a general maximal monotone graph, Adv. Nonlinear Anal., 2017, doi:  https://doi.org/10.1515/anona-2017-0140.
  10. 10.
    Diaz, J.I., Gomez-Castro, D., Podol’skii, A.V., and Shaposhnikova, T.A., Homogenization of boundary value problems in plane domains with frequently alternating type of nonlinear boundary conditions: critical case, Dokl. Math., 2018, vol. 97, no. 3, pp. 271–276.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Diaz, J.I., Gomez-Castro, D., Podolskiy A.V., and Shaposhnikova, T.A., Homogenization of a net of periodic critically scaled boundary obstacles related to reverse osmosis “nano-composite” membranes, Adv. Nonlinear Anal., 2018, doi:  https://doi.org/10.1515/anona-2018-0158.
  12. 12.
    Damlamian, A. and Ta-Tsien, L., Boundary homogenization for elliptic problems, J. Math. Pures Appl., 1987, vol. 66, no. 4, pp. 351–361.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Chechkin, G.A., Averaging of boundary value problems with a singular perturbation of the boundary conditions, Sb. Math., 1994, vol. 79, no. 1, pp. 191–222.MathSciNetGoogle Scholar
  14. 14.
    Borisov, D.I., On a model boundary value problem for Laplacian with frequently alternating type of boundary condition, Asymp. Anal., 2003, vol. 35, no. 1, pp. 1–26.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Zubova, M.N. and Shaposhnikova, T.A., Homogenization of the variational inequality corresponding to a problem with rapidly varying boundary conditions, Math. Notes, 2007, vol. 82, no. 4, pp. 481–489.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Paris: Dunod, 1969. Translated under the title Nekotorye metody resheniya nelineinykh kraevykh zadach, Moscow: Mir, 1972.zbMATHGoogle Scholar
  17. 17.
    Lobo, M., Oleinik, O.A., Perez, M.E., and Shaposhnikova, T.A., On homogenization of solutions of boundary value problems in domains, perforated along manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), 1997, vol. 25, no. 3–4, pp. 611–629.MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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