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 \(B_\varepsilon >0\). Coefficients of the operator \(B_\varepsilon\) are periodic with respect to some lattice in \(\mathbb{R}^d\) and depend on \(\mathbf{x}/\varepsilon\). We study the quantitative homogenization for the solutions of the hyperbolic system \(\partial _t^2\mathbf{u}_\varepsilon =-B_\varepsilon\mathbf{u}_\varepsilon\). In operator terms, we are interested in approximations of the operators \(\cos (tB_\varepsilon ^{1/2})\) and \(B_\varepsilon ^{-1/2}\sin (tB_\varepsilon ^{1/2})\) in suitable operator norms. Approximations for the resolvent \(B_\varepsilon ^{-1}\) have been already obtained by T.A. Suslina. So, we rewrite hyperbolic equation as a system for the vector with components \(\mathbf{u}_\varepsilon \) and \(\partial _t\mathbf{u}_\varepsilon\), 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.

DOI 10.1134/S106192082304012X