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Homogenization of diffusion problems with a nonlinear interfacial resistance

  • Patrizia DonatoEmail author
  • Kim Hang Le Nguyen
Article

Abstract

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 

Keywords

Periodic homogenization Elliptic equations with jump Nonlinear interface conditions 

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Copyright information

© 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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