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 on both 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. Introduction There are many publications devoted to homogenization of differential operators with rapidly oscillating coefficients in bounded domains (see, e.g., [1]–[6]). Similar questions for operators in unbounded domains attracted considerably less attention. On the other hand, during recent years the case of an unbounded domain has been studied intensively. In the series of papers [6]–[13], Birman and Suslina developed a new original technique, which allowed them to prove convergence theorems, to obtain order precise estimates for the rates of convergence, and to construct the first terms in the expansion for the resolvent of a wide class of differential operators with rapidly oscillating coefficients in unbounded domains. It should be emphasized that these results were obtained for the uniform norm, while usually results for bounded domains are formulated in the sense of strong or weak convergence. The approach of Birman and Suslina is based on spectral theory and treats homogenization as a threshold phenomenon. It applies to the operators that admit factorization, and at the same time their coefficients must depend on the fast variable x/ε only; no dependence on the slow variable x is allowed. We should also note the paper [14] by Zhikov, where, by employing another technique, he obtained order precise estimates for the rate of convergence for the resolvent of a scalar operator as well as for the case of the operator of elasticity theory. Again, it was assumed that the coefficients are periodic and depend only on the fast variable. The one-dimensional scalar operators with coefficients depending on both fast and slow variables were studied in [15]–[17]. In [15], the Schrödinger operator with rapidly oscillating and compactly supported potential was considered. The object of study was the phenomenon of a new eigenvalue emerging from the threshold of the continuous spectrum. The case of a periodic operator (independent of the small parameter) perturbed by a rapidly oscillating compactly supported potential with an increasing amplitude was 2000 Mathematics Subject Classification. Primary 35B27.