On homogenization of the first initial-boundary value problem for periodic hyperbolic systems (1807.03634v2)
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 $H4(\mathcal{O};\mathbb{C}n)$ 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 $\mathcal{H}\subset H4(\mathcal{O};\mathbb{C}n)$ to $H1(\mathcal{O};\mathbb{C}n)$ 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 $.