Operator-theoretic approach to homogenization of hyperbolic equations: operator estimates with correctors taken into account

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint strongly elliptic second-order differential operator $\mathcal{A}_\varepsilon$. It is assumed that the coefficients of the operator $\mathcal{A}_\varepsilon$ are periodic and depend on $\mathbf{x} / \varepsilon$, where $\varepsilon >0$. We study the behavior of the operators $\cos ( \mathcal{A}_\varepsilon^{1/2}\tau)$ and $\mathcal{A}_\varepsilon^{-1/2}\sin ( \mathcal{A}_\varepsilon^{1/2}\tau)$ for small $\varepsilon$ and $\tau \in \mathbb{R}$. The results are applied to homogenization of the solutions of the Cauchy problem for the hyperbolic equation $\partial_\tau^2 \mathbf{u}_\varepsilon = - \mathcal{A}_\varepsilon \mathbf{u}_\varepsilon$ with the initial data from a special class. It is shown that, for a fixed $\tau$ and $\varepsilon \to 0$, the solution converges in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ to the solution of the homogenized problem; the error is of order $O(\varepsilon)$. For a fixed $\tau$, we obtain approximation of the solution $\mathbf{u}_\varepsilon(\cdot,\tau)$ in $L_2(\mathbb{R}^d;\mathbb{C}^n)$ with an error $O( \varepsilon^2)$, and also approximation of the solution in the Sobolev space $H^1(\mathbb{R}^d;\mathbb{C}^n)$ with an error $O(\varepsilon)$. In these approximations, correctors are taken into account. The dependence of the errors on the parameter $\tau$ is tracked.