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 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 $(H^1\rightarrow L_2)$-norm for this operator is found. Approximation in the $(H^2\rightarrow H^1)$-operator norm with the correction term taken into account is also established. The results are applied to homogenization for the solutions of the nonhomogeneous hyperbolic equation $\partial ^2_\tau \mathbf{u}_\varepsilon =-\mathcal{A}_\varepsilon \mathbf{u}_\varepsilon +\mathbf{F}$.