Ricerche di Matematica

, Volume 61, Issue 2, pp 231–244 | Cite as

The effective conductivity equation for a highly heterogeneous periodic medium



We consider the homogenization of a conductivity equation for a medium made up of a set \({F_\varepsilon}\) (\({\varepsilon}\) being the size of the period of the medium) of highly conductive vertical fibers surrounded by another material (the matrix) assumed to be a poor conductor. The conductivity coefficients in the fibers behave as \({\frac{1}{\varepsilon^2}}\) while whose of the matrix behave as \({\varepsilon^2}\) . We show that the homogenized problem consists of an equality of the kind u(x) = m(x) f (x) where u denotes the macroscopic temperature, f the source term and m(x) a coefficient given by solving some cell equation.


Homogenization Anisotropy Conductivity Fibers 

Mathematics Subject Classification (2010)

35B27 74QXX 76M50 


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  1. 1.
    Allaire G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Braides A., Briane M., Casado-Diaz J.: Homogenization of non-uniformly bounded periodic diffusion energies in dimension two. Nonlinearity 22(6), 1459–1480 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Briane M., Casado-Diaz J.: Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients. J. Diff. Equ. 845(8), 2038–2054 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Camar-Eddine M., Seppecher P.: Closure of the set of diffusion functionals with respect to the Mosco-convergence. Math. Mod. Meth. Appl. Sci. 12(8), 1153–1176 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Charef, H., Sili, A.: The effective equilibrium law for a highly heterogeneous elastic periodic medium (to appear)Google Scholar
  6. 6.
    Cioranescu D., Damlamian A., Griso G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40, 1585–1620 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Lukkassen D., Nguetseng G., Nnang H., Wall P.: Reiterated homogenization of nonlinear monotone operators in a general deterministic setting. J. Funct. Spaces Appl. 7(2), 121–152 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Nguetseng G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–629 (1989)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Sili A.: A diffusion equation through a highly heterogeneous medium. Appl. Anal. 89, 893–904 (2010)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Sili A.: Homogenization of a nonlinear monotone problem in an anisotropic medium, Math. Models Methods Appl. Sci. 14(3), 329–353 (2004)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Tartar, L.: The general theory of homogenization. A personalized introduction. Lecture Notes of the Unione Matematica Italiana, 7. Springer/UMI, Berlin/Bologna (2009)Google Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2011

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité du Sud Toulon-VarLa Garde cedexFrance

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