Homogenization and Localization with an Interface

We consider the homogenization of a spectral problem for a diffusion equation posed in a singularly perturbed periodic medium. Denoting by ε the period, the diffusion coefficients are scaled as ε². The domain is composed of two periodic medium separated by a planar interface, aligned with the periods. Three different situations arise when ε goes to zero. First, there is a global homogenized problem as if there were no interface. Second, the limit is made of two homogenized problems with a Dirichlet boundary condition on the interface. Third, there is an exponential localization near the interface of the first eigenfunction.