Glasnik Matematicki, Vol. 42, No.2 (2007), 375-388.

QUASILINEAR DIRICHLET PROBLEM IN A PERIODICALLY PERFORATED DOMAIN

L. Boukrim, A. Hakim and T. Mekkaoui

Université Moulay Ismail, F.S.T.E, BP509 Boutalamine Errachidia, Maroc

Université Cadi Ayyad, F.S.T.G, BP 549 Av. Abdelkrim Khattabi Marrakech, Maroc

Université Moulay Ismail, F.S.T.E, BP509 Boutalamine Errachidia, Maroc

Abstract.   Consider the following quasilinear Dirichlet problem (Pε)

    -Δp uε = fε     in   Ωε = Ω \ Tε
uε = 0     on   ∂ Ωε

where 1 < p ≤ 2, (fε) a sequence of functions in Lp'(Ω) such that fε → f in Lp'(Ω) where p' is the conjugate of p, Ω is a bounded domain in RN (N ≥ 2) and Tε the union of inclusions contained in Ω which are ε-periodically distributed. In this paper we give the limit problem and errors estimates for the problem (Pε).

2000 Mathematics Subject Classification.   35B27, 35B40, 35J60.

Key words and phrases.   Homogenization, perforated domain, quasilinear elliptic equations, correctors.

Full text (PDF) (free access)

DOI: 10.3336/gm.42.2.11

References:

  1. H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman, London, 1984.
    MathSciNet

  2. L. Boukrim, A. Hakim and T. Mekkaoui, Limiting behaviour and error estimates for the Dirichlet problem with p-Laplacian in a periodically perfored domain, in preparation.

  3. D. Cioranescu and F. Murat, Un terme étrange venu d'aileurs, Collège de France Seminar, Research Notes in Mathematics, Pitman, London, 1982, No. 60, pp. 98-138, No. 70, pp. 154-178.

  4. D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Univ. Press, 1999.
    MathSciNet

  5. G. Dal Maso, An Introduction to Γ-convergence, Birkhäuser, Boston, 1993.
    MathSciNet

  6. G. Dal Maso and P. Longo, Γ-limits of obstacles, Ann. Mat. Pura Appl. 128 (1981), 1-50.
    MathSciNet     CrossRef

  7. E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale, Rend. Acad. Naz. Lincei 58 (1975), 842-850.
    MathSciNet

  8. R. Glowinski and A. Marroco, Sur l'approximation par éléments finis d'ordre un, et la résolution, par pénalisation -dualité, d'une classe de problémes de Dirichlet non linéaires, RAIRO 9 (1975), 41-76.
    MathSciNet

  9. N. Labani and C. Picard, Homogenization of nonlinear Dirichlet problem in a perfored domain, in "Recent advances in nonlinear elliptic and parabolic problems", Pitman Res. Notes Math. Ser. 208 (1989), 296-305.
    MathSciNet

  10. T. Mekkaoui and C. Picard, Error estimates for the homogenization of a quasilinear Dirichlet problem in a periodically perfored domain, Progress in P.D.E: the Metz Survey 2, M. Chipot Ed., Pitman Res. Notes in Math. Series 296 (1993), 185-193.
    MathSciNet

  11. C. Picard, Analyse limite d'équations variationnelles dans un domaine contenant une grille, Math. Model. Numer. Anal. 21 (1987), 293-326.
    MathSciNet

  12. D. Sandri, Sur l'approximation numérique des écoulements quasi-newtoniens dont la viscosité suit la loi puissance ou la loi de carreau, Math. Model. Numer. Anal. 27 (1993), 131-155.
    MathSciNet

  13. L. Tartar, Cours Peccot au collège de France, mars 1977, non publié.

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