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 ${\mathcal{A}}_{\epsilon }$, $\epsilon >0$. The coefficients of the operator ${\mathcal{A}}_{\epsilon }$ are periodic and depend on $\mathbf{x}/\epsilon$. We study the asymptotic behavior of the operator ${\mathcal{A}}_{\epsilon }^{-1/2}\sin (\tau {\mathcal{A}}_{\epsilon }^{1/2})$, $\tau \in \mathbb{R}$, 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 $O(\epsilon )$. The results are applied to homogenization for the solutions of the hyperbolic equation ${\partial }_{\tau }^2{\mathbf{u}}_{\epsilon }=-{\mathcal{A}}_{\epsilon }{\mathbf{u}}_{\epsilon }+\mathbf{F}$. As examples, we consider the acoustics equation, the system of elasticity, and the model equation of electrodynamics.