On operator error estimates for homogenization of hyperbolic systems with periodic coefficients

Abstract

In L_2(\mathbb{R}^d;\mathbb{C}^n) , we consider a selfadjoint matrix strongly elliptic second order differential operator \mathcal{A}_\varepsilon , \varepsilon > 0 . The coefficients of the operator \mathcal{A}_\varepsilon are periodic and depend on \mathbf{x}/\varepsilon . We study the asymptotic behavior of the operator \mathcal{A}_\varepsilon ^{-1/2}\sin (\tau \mathcal{A}_\varepsilon ^{1/2}) , \tau\in\mathbb{R} , in the small period limit. The principal term of approximation in the (H^1 \to L_2) -norm for this operator is found. Approximation in the (H^2 \to H^1) -operator norm with the correction term taken into account is also established. The error estimates are of the sharp order O(\varepsilon) . The results are applied to homogenization for the solutions of the hyperbolic equation \partial ^2_\tau \mathbf{u}_\varepsilon =-\mathcal{A}_\varepsilon \mathbf{u}_\varepsilon +\mathbf{F} . As examples, we consider the acoustics equation, the system of elasticity, and the model equation of electrodynamics.