An Operator-Asymptotic Approach to Periodic Homogenization for Equations of Linearized Elasticity

We present an operator-asymptotic approach to the problem of homogenization of periodic composite media in the setting of three-dimensional linearized elasticity. This is based on a uniform approximation with respect to the inverse wavelength |χ| for the solution to the resolvent problem when written as a superposition of elementary plane waves with wave vector (“quasimomentum”) χ . We develop an asymptotic procedure in powers of |χ|, combined with a new uniform version of the classical Korn inequality. As a consequence, we obtain L² → L², L² → H¹ , and higher-order L² → L² norm-resolvent estimates in ℝ 3. The L² → H¹ and higher-order L² → L² correctors emerge naturally from the asymptotic procedure, and the former is shown to coincide with the classical formulae.