Homogenization of Hyperbolic Equations

We consider a self-adjoint matrix elliptic operator $A_\varepsilon$, $\varepsilon >0$, on $L_2({\mathbb R}^d;{\mathbb C}^n)$ given by the differential expression $b({\mathbf D})^* g({\mathbf x}/\varepsilon)b({\mathbf D})$. The matrix-valued function $g({\mathbf x})$ is bounded, positive definite, and periodic with respect to some lattice; $b({\mathbf D})$ is an $(m\times n)$-matrix first order differential operator such that $m \ge n$ and the symbol $b(\boldsymbol{\xi})$ has maximal rank. We study the operator cosine $\cos (\tau A^{1/2}_\varepsilon)$, where $\tau \in {\mathbb R}$. It is shown that, as $\varepsilon \to 0$, the operator $\cos (\tau A^{1/2}_\varepsilon)$ converges to $\cos(\tau (A^0)^{1/2})$ in the norm of operators acting from the Sobolev space $H^s({\mathbb R}^d;{\mathbb C}^n)$ (with a suitable $s$) to $L_2({\mathbb R}^d;{\mathbb C}^n)$. Here $A^0$ is the effective operator with constant coefficients. Sharp-order error estimates are obtained. The question about the sharpness of the result with respect to the type of the operator norm is studied. Similar results are obtained for more general operators. The results are applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation $\partial^2_\tau {\mathbf u}_\varepsilon ({\mathbf x}, \tau) =- A_\varepsilon {\mathbf u}_\varepsilon({\mathbf x}, \tau)$.