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 ${\mathcal{A}}_{\epsilon }$ with periodic coefficients depending on $\mathbf{x}/\epsilon$, $\epsilon >0$. We study the behavior of the operators $\cos ({\mathcal{A}}_{\epsilon }^{1/2}\tau )$ and ${\mathcal{A}}_{\epsilon }^{-1/2}\sin ({\mathcal{A}}_{\epsilon }^{1/2}\tau )$, $\tau \in \mathbb{R}$, for small $\epsilon$. 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 ${\mathbf{v}}_{\epsilon }$ of the Cauchy problem for the hyperbolic equation ${\partial }_{\tau }^2{\mathbf{v}}_{\epsilon }=-{\mathcal{A}}_{\epsilon }{\mathbf{v}}_{\epsilon }+\mathbf{F}$. General results are applied to the acoustics equation and the system of elasticity theory.

41 pages. arXiv admin note: substantial text overlap with arXiv:1606.05868, arXiv:1508.07641