Geodesic
The Limit Absorption Principle and Homogenization Procedure for Periodic Elliptic Operators
Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 4, pp. 105-108.

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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 $0\sigma1/2$.
Keywords: periodic differential operators, homogenization, effective operator, limit absorption principle. 
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M. S. Birman; T. A. Suslina. The Limit Absorption Principle and Homogenization Procedure for Periodic Elliptic Operators. Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 4, pp. 105-108. http://geodesic.mathdoc.fr/item/FAA_2008_42_4_a8/

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