On homogenization of the first initial-boundary value problem for periodic hyperbolic systems

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 for 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 $.