Effective Transmission Conditions for Reaction-Diffusion Processes in Domains Separated by an Interface

Abstract

In this paper, we develop multiscale methods appropriate for the homogenization of processes in domains containing thin heterogeneous layers. Our model problem consists of a nonlinear reaction-diffusion system defined in such a domain, and properly scaled in the layer region. Both the period of the heterogeneities and the thickness of the layer are of order $\varepsilon.$ By performing an asymptotic analysis with respect to the scale parameter $\varepsilon$ we derive an effective model which consists of the reaction-diffusion equations on two domains separated by an interface together with appropriate transmission conditions across this interface. These conditions are determined by solving local problems on the standard periodicity cell in the layer. Our asymptotic analysis is based on weak and strong two-scale convergence results for sequences of functions defined on thin heterogeneous layers. For the derivation of the transmission conditions, we develop a new method based on test functions of boundary l...