Asymptotics for the solutions of elliptic systems with rapidly oscillating coefficients

A singularly perturbed second order elliptic system in the entire space is treated. The coefficients of the systems oscillate rapidly and depend both on the slow and fast variables. The homogenized operator is obtained and, in the uniform norm sense, the leading terms of the asymptotic expansion are constructed for the resolvent of the operator described by the system. The convergence of the spectrum is established, and examples are given.