Multiscale analysis of a diffusion problem in a composite material with a rough imperfect interface

In this paper, we consider a stationary heat diffusion problem in a two-component material, which exhibits an imperfect contact across a rapidly oscillating interface separating the two constituents. At the microscopic scale, its mathematical description is given by an elliptic boundary value problem stated in a domain made up of two connected components separated by the oscillating interface. Across this interface, we assume the continuity of the heat potential and we prescribe a jump condition for its flux, given by an oscillating source, modulated by a factor depending on the length of the microscale. Our problem involves also two parameters, which are related to the amplitude of the oscillations of the rough interface and to the magnitude of the source term on this interface, respectively. We perform an asymptotic analysis with the aim of deriving a limit model, which can describe the macroscopic behavior of the material under investigation. We prove that only for three combined choices of the above mentioned parameters the influence of the source and of the interface geometry is kept in the limit, producing three different asymptotic models, all involving an imperfect flat interface.