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 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 results are applied to homogenization for the solutions of the nonhomogeneous hyperbolic equation ${\partial }_{\tau }^2{\mathbf{u}}_{\epsilon }=-{\mathcal{A}}_{\epsilon }{\mathbf{u}}_{\epsilon }+\mathbf{F}$.

arXiv commentMajor revision of the third version. Some inaccuracies have been corrected. The presentation became more compact. New material added. Some applications of the general results were considered (see Section 11). 42 pages