Homogenization of diffusion problems with a nonlinear interfacial resistance

  • Patrizia DonatoEmail author
  • Kim Hang Le Nguyen


In this paper, we consider a stationary heat problem on a two-component domain with an ε-periodic imperfect interface, on which the heat flux is proportional via a nonlinear function to the jump of the solution, and depends on a real parameter γ. Homogenization and corrector results for the corresponding linear case have been proved in Donato et al. (J Math Sci 176(6):891–927, 2011), by adapting the periodic unfolding method [see (Cioranescu et al. SIAM J Math Anal 40(4):1585–1620, 2008), (Cioranescu et al. SIAM J Math Anal 44(2):718–760, 2012), (Cioranescu et al. Asymptot Anal 53(4):209–235, 2007)] to the case of a two-component domain. Here, we first prove, under natural growth assumptions on the nonlinearities, the existence and the uniqueness of a solution of the problem. Then, we study, using the periodic unfolding method, its asymptotic behavior when \({\varepsilon\to 0}\). In order to describe the homogenized problem, we complete some convergence results of Donato et al. (J Math Sci 176(6):891–927, 2011) concerning the unfolding operators and we investigate the limit behaviour of the unfolded Nemytskii operators associated to the nonlinear terms. According to the values of the parameter γ we have different limit problems, for the cases \({\gamma < -1, \gamma =-1}\) and \({\gamma \in \left] -1,1\right]}\). The most relevant case is \({\gamma =-1}\), where the homogenized matrix differs from that of the linear case, and is described in a more complicated way, via a nonlinear function involving the correctors.

Mathematics Subject Classification

35B27 35J65 82B24 


Periodic homogenization Elliptic equations with jump Nonlinear interface conditions 


  1. 1.
    Allaire G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Amar M., Andreucci D., Gianni R.: Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues. Math. Model. Methods Appl. Sci. 14, 1261–1295 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Amar M., Andreucci D., Bisegna P., Gianni R.: A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case. Differ. Integral Equations 26(9–10), 885–912 (2013)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Auriault J.L., Ene H.: Macroscopic modelling of heat transfer in composites with interfacial thermal barrier. Int. J. Heat Mass Tranfer 37, 2885–2892 (1994)zbMATHCrossRefGoogle Scholar
  5. 5.
    Barenblatt G.I., Zheltov Y.P., Kochina I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fractured rocks. Prikl. Mat. Mekh. 24(5), 852–864 (1960)Google Scholar
  6. 6.
    Bensoussan, A.; Lions, J.-L.; Papanicolaou, G: Asymptotic Analysis for Periodic Structures. North Holland, Amsterdam (1978)Google Scholar
  7. 7.
    Cabarrubias B., Donato P.: Existence and uniqueness for a quasilinear elliptic problem with nonlinear Robin conditions. Carpathian J. Math. 27(2), 173–184 (2011)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Cabarrubias B., Donato P.: Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions. Appl. Anal. 91(6), 1111–1127 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Canon E., Pernin J.N.: Homogénéisation d’un problème de diffusion en milieu composite avec barrière à à l’interface. C. R. Acad. Sci., Ser. I, Math. 325(1), 123–126 (1997)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Carslaw H.S., Jaeger J.C.: Conduction of Heat in Solids. At the Clarendon Press, Oxford (1959)Google Scholar
  11. 11.
    Cioranescu D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C. R. Acad. Sci. Paris, Série 1, 335, 99–104 (2002)Google Scholar
  12. 12.
    Cioranescu D., Damlamian A., Griso G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40(4), 1585–1620 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Cioranescu D., Damlamian A., Donato P., Griso G., Zaki R.: The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44(2), 718–760 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Cioranescu, D., Donato, P.: An Introduction to Homogenization, Oxford Lecture Series in Mathematics and Its Applications, vol. 17 (1999)Google Scholar
  15. 15.
    Cioranescu D., Donato P., Zaki R.: The periodic unfolding method in perforated domains. Portugaliae Math. 63(4), 467–496 (2006)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Cioranescu D., Donato P., Zaki R.: Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions. Asymptot. Anal. 53(4), 209–235 (2007)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Cioranescu D., Saint Jean Paulin J.: Homogenization open sets with holes. J. Math. Anal. Appl. 71, 590–607 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Cioranescu, D., Paulin, J.S.J.: Homogenization of reticulated structures. In: Appl. Math. Sci., vol. 139. Springer, New York (1999)Google Scholar
  19. 19.
    Damlamian A., Meunier N., Van Schaftingen J.: Periodic homogenization of monotone multivalued operators. Nonlinear Anal. 67, 3217–3239 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Donato P.: Some corrector results for composites with imperfect interface. Rend. Math. Ser. VII 26, 189–209 (2006)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Donato, P.: Homogenization of a class of imperfect transmission problems. In: Damlamian, A., Miara, B., Li, T. (eds.) Multiscale Problems: Theory, Numerical Approximation and Applications. Series in Contemporary Applied Mathematics CAM, vol. 16, pp. 109–147. Higher Education Press, Beijing (2011)Google Scholar
  22. 22.
    Donato P., Faella L., Monsurrò S.: Homogenization of the wave equation in composites with imperfect interface: a memory effect. J. Math. Pures Appl. 87, 119–143 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Donato P., Faella L., Monsurrò S.: Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces. SIAM J. Math. Anal. 40(5), 1952–1978 (2009)zbMATHCrossRefGoogle Scholar
  24. 24.
    Donato P., Jose E.C.: Corrector results for a parabolic problem with a memory effect. ESAIM Math. Model. Numerical Anal. 44, 421–454 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Donato P., Monsurrò S.: Homogenization of two heat conductors with an interfacial contact resistance. Anal. Appl. 2(3), 247–273 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Donato P., Le Nguyen K.H., Tardieu R.: The periodic unfolding method for a class of imperfect transmission problems. J. Math. Sci. 176(6), 891–927 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Ene H.I.: On the microstructure models of porous media. Rev. Roum. Math. Pures Appl. 46(2–3), 289–295 (2001)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Ene H.I., Poliševski D.: Model of diffusion in partially fissured media. ZAMP 53, 1052–1059 (2002)zbMATHCrossRefGoogle Scholar
  29. 29.
    Faella, L., Monsurrò, S: Memory effects arising in the homogenization of composites with inclusions. In: Topics on Mathematics for Smart Systems, pp. 107–121. World Sci. Publ, Hackensack (2007)Google Scholar
  30. 30.
    Hummel H.K.: Homogenization for heat tranfer in polycrystals with interfacial resistances. Appl. Anal. 75(3–4), 403–424 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Jose E.C.: Homogenization of a parabolic problem with an imperfect interface. Rev. Roum. Math. Pures Appl. 54(3), 189–222 (2009)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Krasnosel’skii, MA.: Topological Methods in the theory of nonlinear intergral equations. In: International Series of Monographs in Pure and Applied Mathematics. Pergamon Press, New York (1964)Google Scholar
  33. 33.
    Lipton R.: Heat conduction in fine scale mixtures with interfacial contact resistance. Siam J. Appl. Math. 58(1), 55–72 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Lipton R., Vernescu B.: Composite with imperfect interface. Proc. Soc. Lond. A 452, 329–358 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Mardare, S.: Personal communicationGoogle Scholar
  36. 36.
    Monsurrò S.: Homogenization of a two-component composite with interfacial thermal barrier. Adv. Math. Sci. Appl. 13(1), 43–63 (2003)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Monsurrò S.: Erratum for the paper homogenization of a two-component composite with interfacial thermal barrier. Adv. Math. Sci. Appl. 14, 375–377 (2004)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Nguetseng G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20, 608–629 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    Pernin, J.N.: Homogénéisation d’un problème de diffusion en milieu composite à à deux composantes. C.R. Acad. Sci. Paris Série I, 321, 949–952 (1995)Google Scholar
  40. 40.
    Tartar, L.: Quelques remarques sur l’homogénéisation. In: Functional Analysis and Numerical Analysis, Proc. Japan–France Seminar 1976 (Fujita ed.), Japanese Society for the Promotion of Science, pp. 468–482 (1978)Google Scholar
  41. 41.
    Timofte, C.: Upscaling in nonlinear diffusion problems in composite materials, Progress in Industrial Mathematics at ECMI (2006), 328–332Google Scholar

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Raphaël Salem, CNRS UMR 6085Normandie Université, Université de RouenSaint-Étienne du Rouvray CedexFrance
  2. 2.Faculty of SciencesNong Lam UniversityHo Chi Minh CityVietnam

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