Variations on the theme of the Trotter-Kato theorem for homogenization of periodic hyperbolic systems

Abstract

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, we consider a matrix elliptic second order differential operator $B_{\epsilon }>0$. Coefficients of the operator $B_{\epsilon }$ are periodic with respect to some lattice in ${\mathbb{R}}^d$ and depend on $\mathbf{x}/\epsilon$. We study the quantitative homogenization for the solutions of the hyperbolic system ${\partial }_t^2{\mathbf{u}}_{\epsilon }=-B_{\epsilon }{\mathbf{u}}_{\epsilon }$. In operator terms, we are interested in approximations of the operators $\cos (t{B}_{\epsilon }^{1/2})$ and $B_{\epsilon }^{-1/2}\sin (t{B}_{\epsilon }^{1/2})$ in suitable operator norms. Approximations for the resolvent $B_{\epsilon }^{-1}$ have been already obtained by T.~A.~Suslina. So, we rewrite hyperbolic equation as a system for the vector with components ${\mathbf{u}}_{\epsilon }$ and ${\partial }_t{\mathbf{u}}_{\epsilon }$, and consider the corresponding unitary group. For this group, we adapt the proof of the Trotter-Kato theorem by introduction of some correction term and derive hyperbolic results from elliptic ones.

62 pages. Version 3 is a minor revision of version 2. Some of notation were changed. Version 2 is a minor revision of version 1. Some typos have been corrected. Inaccuracies in Theorem 3.13 and in the proof of Theorem 4.9 have been fixed

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• Homogenization of the Dirichlet problem for elliptic systems: Two-parametric error estimates
• On homogenization of the first initial-boundary value problem for periodic hyperbolic systems
• Homogenization of hyperbolic equations with periodic coefficients in ${\mathbb{R}}^d$: sharpness of the results
• Weak and strong approximation of semigroups on Hilbert spaces