[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