Spectral approach to homogenization of hyperbolic equations with periodic coefficients

TL;DR

This paper develops a spectral method to analyze the homogenization of hyperbolic equations with periodic coefficients, providing operator approximations and applying results to acoustics and elasticity systems.

Contribution

It introduces a spectral approach to homogenize hyperbolic equations with periodic coefficients, deriving operator approximations and analyzing solutions in this context.

Findings

Operator approximations in the $(H^s\to L_2)$-norm for small $\varepsilon$

Effective behavior of solutions to hyperbolic equations with periodic coefficients

Application to acoustics and elasticity systems

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.