Homogenization of the first initial-boundary value problem for periodic hyperbolic systems. Principal term of approximation

Abstract

Let $\mathcal{O}\subset {\mathbb{R}}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};{\mathbb{C}}^n)$, we consider a matrix elliptic second order differential operator $A_{D,\epsilon }$ with the Dirichlet boundary condition. Here $\epsilon >0$ is a small parameter. The coefficients of the operator $A_{D,\epsilon }$ are periodic and depend on $\mathbf{x}/\epsilon$. The principal terms of approximations for the operator cosine and sine functions are given in the $(H^2\to L_2)$- and $(H^1\to L_2)$-operator norms, respectively. The error estimates are of the precise order $O(\epsilon )$ for a fixed time. The results in operator terms are derived from the quantitative homogenization estimate for approximation of the solution of the initial-boundary value problem for the equation $({\partial }_t^2+A_{D,\epsilon }){\mathbf{u}}_{\epsilon }=\mathbf{F}$.

17 pages. One inaccuracy removed