[Authors]
 Meshkova, Yu. M.

[Title]
 On operator estimates for homogenization of hyperbolic systems with periodic
 coefficients

[AMS Subj-class]
 35B27 Homogenization; equations in media with periodic structure
 35L52 Initial value problems for second-order hyperbolic systems

[Keywords]
 periodic differential operators, hyperbolic systems, homogenization, operator
 error estimates

[Abstract]
 In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, selfadjoint matrix strongly elliptic
 differential operators $\mathcal{A}_\varepsilon$, $\varepsilon >0$, are
 considered. 
 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
 $\partial ^2_\tau \mathbf{u}_\varepsilon =-\mathcal{A}_\varepsilon
 \mathbf{u}_\varepsilon +\mathbf{F}$.

[Comments]
 LaTeX, Russian, 35 pp.
 Postal address: 
 Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29B,
 Saint Petersburg 199178 Russia.

[Contact e-mail]
 juliavmeshke@yandex.ru