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 $\mathcal {A}_\varepsilon$ is considered. It is assumed that the coefficients of $\mathcal {A}_\varepsilon$ are periodic and depend on $\mathbf {x}/\varepsilon$, where $\varepsilon >0$ is a small parameter. We find approximations for the operators $\cos (\mathcal {A}_\varepsilon ^{1/2}\tau )$ and $\mathcal {A}_\varepsilon ^{-1/2}\sin (\mathcal {A}_\varepsilon ^{1/2}\tau )$ 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 $\mathcal {A}_\varepsilon ^{-1/2}\sin (\mathcal {A}_\varepsilon ^{1/2}\tau )$ 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 $\partial _\tau ^2 \mathbf {u}_\varepsilon = -\mathcal {A}_\varepsilon \mathbf {u}_\varepsilon + \mathbf {F}$.