The Limit Absorption Principle and Homogenization Procedure for Periodic Elliptic Operators

For a periodic matrix elliptic operator $\mathcal{A}_\varepsilon$ with (${\mathbf x}/\varepsilon$-dependent) rapidly oscillating coefficients, a certain analog of the limit absorption principle is proved. It is shown that the bordered resolvent $\langle{\mathbf x}\rangle^{-1/2-\delta}(\mathcal{A}_\varepsilon-(\eta\pm i\varepsilon^\sigma)I)^{-1}\langle{\mathbf x}\rangle^{-1/2-\delta}$ has a limit in the operator norm in $L_2$ as $\varepsilon\to 0$ provided that $\eta>0$, $\delta>0$, and $0<\sigma<1/2$.