[Authors]
 Meshkova, Yu. M.

[Title]
 On homogenization of the first initial-boundary value problem for periodic
 hyperbolic systems via the inverse Laplace transform

[AMS Subj-class]
 35B27 Homogenization; equations in media with periodic structure
 35L53 Initial-boundary value problems for second-order hyperbolic systems

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

[Abstract]
 Let $\mathcal{O}\subset\mathbb{R}^d$ a bounded domain of class  $C^{1,1}$. 
 In $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a self-adjoint matrix strongly
 elliptic second order differential operator  $B_{D,\varepsilon}$,
 $0<\varepsilon \leqslant 1$, with the Dirichlet boundary condition. 
 The coefficients of the operator $B_{D,\varepsilon}$ are periodic and depend
 on $\mathbf{x}/\varepsilon$. We are interested in the behavior of the operators
 $\cos(tB_{D,\varepsilon}^{1/2})$ and
 $B_{D,\varepsilon} ^{-1/2}\sin (t B_{D,\varepsilon} ^{1/2})$, $t\in\mathbb{R}$,
 in the small period limit. For these operators, approximations in the norm of
 operators acting from some subspace $\mathcal{H}$ of the Sobolev space
 $H^4(\mathcal{O};\mathbb{C}^n)$ to $L_2(\mathcal{O};\mathbb{C}^n)$ are found. 
 Moreover, for $B_{D,\varepsilon} ^{-1/2}\sin (t B_{D,\varepsilon} ^{1/2})$,
 the approximation with the corrector in the norm of operators acting from
 $\mathcal{H}\subset H^4(\mathcal{O};\mathbb{C}^n)$ to
 $H^1(\mathcal{O};\mathbb{C}^n)$ is obtained.  The results are applied to
 homogenization of the solution of the first initial-boundary value problem
 for the hyperbolic equation 
 $\partial ^2_t \mathbf{u}_\varepsilon =-B_{D,\varepsilon} \mathbf{u}_\varepsilon$.

[Comments]
 LaTeX, Russian, 36 pp.
 Postal address: 
 Chebyshev Laboratory, St. Petersburg State University, 14th Line V.O., 29B,
 Saint Petersburg 199178 Russia.

[Contact e-mail]
 y.meshkova@spbu.ru