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

### Yulia M. Meshkova

St. Petersburg State University, Russia

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## 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.

## Cite this article

Yulia M. Meshkova, On operator error estimates for homogenization of hyperbolic systems with periodic coefficients. J. Spectr. Theory 11 (2021), no. 2, pp. 587–660

DOI 10.4171/JST/350