The limit absorption principle and homogenization procedure for periodic elliptic operators

Abstract

For a periodic matrix elliptic operator \( A_\varepsilon \) with (xɛ-dependent) rapidly oscillating coefficients, a certain analog of the limit absorption principle is proved. It is shown that the bordered resolvent 〈x〉−1/2−·(\( A_\varepsilon \) − (η ± iɛσ)I)−1〈x〉−1/2−· has a limit in the operator norm in L2 as ɛ → 0 provided that η > 0, · > 0, and 0 < σ < 1/2.