Homogenization and localization for a 1D eigenvalue problem in a periodic medium with an interface (Q2568744)

This paper deals with the homogenization of the eigenvalue problem for a singularly perturbed ordinary differential equation of second order (one-dimensional diffusion equation). Denoting by \(\varepsilon\) the period, the diffusion coefficient is scaled as \(\varepsilon^2\). The domain is made of two purely periodic media separated by a point boundary. The authors introduce a discontinuity coefficient \(\alpha\) determined by the interface condition. For instance, the case \(\alpha=0\) is reduced to the purely periodic case. If \(\alpha>0\), each eigenfunction goes to \(0\) at the interface. If \(\alpha<0\), the first eigenfunction concentrates exponentially at the interface. Although the authors discuss only the one-dimensional equation, even the statement of the problem is new in the theory of homogenization.