Spectral approach to homogenization of hyperbolic equations with periodic coefficients

Abstract: In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider selfadjoint strongly elliptic second order differential operators ${\mathcal A}\varepsilon$ with periodic coefficients depending on ${\mathbf x}/ \varepsilon$, $\varepsilon>0$. We study the behavior of the operators $\cos( {\mathcal A}^{1/2}\varepsilon \tau)$ and ${\mathcal A}^{-1/2}_\varepsilon \sin( {\mathcal A}^{1/2}_\varepsilon \tau)$, $\tau \in \mathbb{R}$, for small $\varepsilon$. Approximations for these operators in the $(H^s\to L_2)$-operator norm with a suitable $s$ are obtained. The results are used to study the behavior of the solution ${\mathbf v}\varepsilon$ of the Cauchy problem for the hyperbolic equation $\partial^2\tau {\mathbf v}\varepsilon = - \mathcal{A}\varepsilon {\mathbf v}_\varepsilon +\mathbf{F}$. General results are applied to the acoustics equation and the system of elasticity theory.