Homogenization of diffusion problems with a nonlinear interfacial resistance

International audienceIn this paper, we consider a stationary heat problem on a two-component domain with an epsilon-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 gamma. 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 . 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 gamma we have different limit problems, for the cases and . The most relevant case is , 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