Homogenization of hyperbolic equations with periodic coefficients in ℝ^{𝕕}: Sharpness of the results

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