Homogenization of diffusion problems with a nonlinear interfacial resistance

• Published in: Nonlinear differential equations and applications 2015-10, Vol.22 (5), p.1345-1380
• Main Authors: Donato, Patrizia, Le Nguyen, Kim Hang
• Format: Article
• Language: English
• Subjects: Analysis; Analysis of PDEs; Mathematics; Mathematics and Statistics
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-015-0325-2

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 ε → 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 γ < - 1 , γ = - 1 and γ ∈ - 1 , 1 . The most relevant case is γ = - 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.