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We present an operator-asymptotic approach to the problem of homogenizationof periodic composite media in the setting of three-dimensional linearizedelasticity. This is based on a uniform approximation with respect to theinverse wavelength $|\chi|$ for the solution to the resolvent problem whenwritten as a superposition of elementary plane waves with wave vector(``quasimomentum") $\chi$. We develop an asymptotic procedure in powers of$|\chi|$, combined with a new uniform version of the classical Korn inequality.As a consequence, we obtain $L^2\to L^2$, $L^2\to H^1$, and higher-order$L^2\to L^2$ norm-resolvent estimates in $\mathbb{R}^3$. The $L^2 \to H^1$ andhigher-order $L^2 \to L^2$ correctors emerge naturally from the asymptoticprocedure, and the former is shown to coincide with the classical formulae.

An operator-asymptotic approach to periodic homogenization for equations of linearized elasticity