Applications of Mathematics

, Volume 54, Issue 6, pp 465–489 | Cite as

Σ-convergence of nonlinear monotone operators in perforated domains with holes of small size

  • Jean Louis WoukengEmail author


This paper is devoted to the homogenization beyond the periodic setting, of nonlinear monotone operators in a domain in ℝ N with isolated holes of size ɛ2 (ɛ > 0 a small parameter). The order of the size of the holes is twice that of the oscillations of the coefficients of the operator, so that the problem under consideration is a reiterated homogenization problem in perforated domains. The usual periodic perforation of the domain and the classical periodicity hypothesis on the coefficients of the operator are here replaced by an abstract assumption covering a great variety of behaviors such as the periodicity, the almost periodicity and many more besides. We illustrate this abstract setting by working out a few concrete homogenization problems. Our main tool is the recent theory of homogenization structures.


perforated domains homogenization reiterated 

MSC 2000

35B40 46J10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Allaire, F. Murat: Homogenization of the Neumann problem with nonisolated holes. Asymptotic Anal. 7 (1993), 81–95.zbMATHMathSciNetGoogle Scholar
  2. [2]
    B. Amaziane, M. Goncharenko, L. Pankratov: Homogenization of a convection-diffusion equation in perforated domains with a weak adsorption. Z. Angew. Math. Phys. 58 (2007), 592–611.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. S. Besicovitch: Almost Periodic Functions. Dover Publications, New York, 1955.zbMATHGoogle Scholar
  4. [4]
    N. Bourbaki: Éléments de mathématique. Intégration, Chap. 1–4. Hermann, Paris, 1966. (In French.)Google Scholar
  5. [5]
    N. Bourbaki: Éléments de matématique. Topologie générale, Chap. 1–4. Hermann, Paris, 1971. (In French.)Google Scholar
  6. [6]
    G. Cardone, P. Donato, A. Gaudiello: A compactness result for elliptic equations with subquadratic growth in perforated domains. Nonlin. Anal., Theory Methods Appl. 32 (1998), 335–361.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    V. Chiadò Piat, A. Defranceschi: Asymptotic behaviour of quasi-linear problems with Neumann boundary conditions on perforated domains. Appl. Anal. 36 (1990), 65–87.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    D. Cioranescu, J. Saint Jean Paulin: Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979), 590–607.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    D. Cioranescu, P. Donato: An introduction to homogenization. Oxford Lecture Series in Mathematics and its Applications, 17. Oxford University Press, Oxford, 1999.zbMATHGoogle Scholar
  10. [10]
    D. Cioranescu, F. Murat: Un terme étrange venu d’ailleurs. Nonlin. partial differential equations and their applications, Coll. de France Semin., Vol. II, Vol. III. Res. Notes Math. (1982), 98–138, 154–178. (In French.)Google Scholar
  11. [11]
    C. Conca, P. Donato: Non-homogeneous Neumann problems in domains with small holes. RAIRO, Modél. Math. Anal. Numér. 22 (1988), 561–607.zbMATHMathSciNetGoogle Scholar
  12. [12]
    P. Donato, L. Sgambati: Homogenization for some nonlinear problems in perforated domains. Rev. Mat. Apl. 15 (1994), 17–38.zbMATHMathSciNetGoogle Scholar
  13. [13]
    J. F. F. Fournier, J. Stewart: Amalgams of Lp and lq. Bull. Am. Math. Soc. 13 (1985), 1–21.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    A. Guichardet: Analyse harmonique commutative. Dunod, Paris, 1968. (In French.)zbMATHGoogle Scholar
  15. [15]
    C. Huang: Homogenization of biharmonic equations in domains perforated with tiny holes. Asymptotic Anal. 15 (1997), 203–227.zbMATHGoogle Scholar
  16. [16]
    R. Larsen: Banach Algebras. An Introduction. Marcel Dekker, New York, 1973.zbMATHGoogle Scholar
  17. [17]
    J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod/Gauthier-Vilars, Paris, 1969. (In French.)zbMATHGoogle Scholar
  18. [18]
    D. Lukkassen, G. Nguetseng, H. Nnang, P. Wall: Reiterated homogenization of nonlinear elliptic operators in a general deterministic setting. J. Funct. Spaces Appl. 7 (2009), 121–152.zbMATHMathSciNetGoogle Scholar
  19. [19]
    G. Nguetseng: Homogenization structures and applications I. Z. Anal. Anwend. 22 (2003), 73–107.zbMATHMathSciNetGoogle Scholar
  20. [20]
    G. Nguetseng: Homogenization structures and applications II. Z. Anal. Anwend. 23 (2004), 483–508.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    G. Nguetseng: Mean value on locally compact abelian groups. Acta Sci. Math. 69 (2003), 203–221.zbMATHMathSciNetGoogle Scholar
  22. [22]
    G. Nguetseng: Almost periodic homogenization Asymptotic analysis of a second order elliptic equation. Preprint.Google Scholar
  23. [23]
    G. Nguetseng: Homogenization in perforated domains beyond the periodic setting. J. Math. Anal. Appl. 289 (2004), 608–628.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    G. Nguetseng, H. Nnang: Homogenization of nonlinear monotone operators beyond the periodic setting. Electron. J. Differ. Equ. 2003 (2003).Google Scholar
  25. [25]
    G. Nguetseng, J. L. Woukeng: Deterministic homogenization of parabolic monotone operators with time dependent coefficients. Electron. J. Differ. Equ. 2004 (2004).Google Scholar
  26. [26]
    G. Nguetseng, J. L. Woukeng: Σ-convergence of nonlinear parabolic operators. Nonlinear Anal., Theory Methods Appl. 66 (2007), 968–1004.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    O.A. Oleinik, T.A. Shaposhnikova: On the biharmonic equation in a domain perforated along manifolds of small dimension. Differ. Uravn. 32 (1996), 335–362.MathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2009

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of DschangDschangCameroon

Personalised recommendations