Homogenization of hyperbolic equations with periodic coefficients in ${\mathbb{R}}^d$: sharpness of the results

Abstract

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, a selfadjoint strongly elliptic second order differential operator ${\mathcal{A}}_{\epsilon }$ is considered. It is assumed that the coefficients of the operator ${\mathcal{A}}_{\epsilon }$ are periodic and depend on $\mathbf{x}/\epsilon$, where $\epsilon >0$ is a small parameter. We find approximations for the operators $\cos ({\mathcal{A}}_{\epsilon }^{1/2}\tau )$ and ${\mathcal{A}}_{\epsilon }^{-1/2}\sin ({\mathcal{A}}_{\epsilon }^{1/2}\tau )$ in the norm of operators acting from the Sobolev space $H^s({\mathbb{R}}^d)$ to $L_2({\mathbb{R}}^d)$ (with suitable $s$). We also find approximation with corrector for the operator ${\mathcal{A}}_{\epsilon }^{-1/2}\sin ({\mathcal{A}}_{\epsilon }^{1/2}\tau )$ in the $(H^s\to H^1)$-norm. The question about the sharpness of the results with respect to the type of the operator norm and with respect to the dependence of estimates on $\tau$ is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation ${\partial }_{\tau }^2{\mathbf{u}}_{\epsilon }=-{\mathcal{A}}_{\epsilon }{\mathbf{u}}_{\epsilon }+\mathbf{F}$.

95 pages