Homogenization of Hyperbolic Equations with Periodic Coefficients

The talk is based on joint works with Mark Dorodnyi. I will give a survey of the results [1], [2], [3], [4], [5] on homogenization of hyperbolic equations in $\mathbb{R}^d$.
In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a strongly elliptic differential operator (DO) $A_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$, $\varepsilon >0$. Here $g(\mathbf{x})$ is a bounded and positive definite periodic $(m \times m)$-matrix-valued function; $b(\mathbf{D}) = \sum_{l=1}^d b_l D_l$ is an $(m\times n)$-matrix first-order DO. It is assumed that $m \geqslant n$ and the symbol $b(\boldsymbol{\xi})$ has maximal rank. We are interested in the behavior of the operators $\cos (\tau A_\varepsilon^{1/2})$ and $A_\varepsilon^{-1/2} \sin (\tau A_\varepsilon^{1/2})$.
As $\varepsilon \to 0$, these operators converge in the norm of operators acting from the Sobolev space $H^s(\mathbb{R}^d;\mathbb{C}^n)$ to $L_2(\mathbb{R}^d;\mathbb{C}^n)$ (with suitable $s$) to the similar operator-valued functions of the effective operator $A^0$, the error being of order $O(\varepsilon)$ for a fixed $\tau \in {\mathbb R}$. We will also discuss the possibility to obtain more accurate approximations in the $(H^s \to L_2)$-norm with error of order $O(\varepsilon^2)$, as well as approximations in the $(H^s \to H^1)$-norm with error of order $O(\varepsilon)$. It turns out that such approximations (with appropriate corrector terms) can be obtained for the operators $A_\varepsilon^{-1/2} \sin (\tau A_\varepsilon^{1/2})$ and $\cos (\tau A_\varepsilon^{1/2}) \left(I + \varepsilon K(\varepsilon) \right)$, where $K(\varepsilon) = \Lambda^\varepsilon b(\mathbf{D}) \Pi_\varepsilon$ involves a rapidly oscillating coefficient $\Lambda^\varepsilon(\mathbf{x}) = \Lambda(\mathbf{x}/\varepsilon)$ and an auxiliary smoothing operator $\Pi_\varepsilon$.
A special attention is paid to the sharpness of the results. The results are applied to the Cauchy problem for a hyperbolic equation $\partial_\tau^2 {\mathbf u}_\varepsilon = - A_\varepsilon {\mathbf u}_\varepsilon + {\mathbf F}$.
[1] M.Sh. Birman and T.A. Suslina, Operator error estimates in the homogenization problem for nonstationary periodic equations, St. Petersburg Math. J. 20 (2009), no. 6, 873–928.
[2] Yu.M. Meshkova, On operator error estimates for homogenization of hyperbolic systems with periodic coeffcients, J. Spectr. Theory 11 (2021), no. 2, 587–660.
[3] M.A. Dorodnyi and T.A. Suslina, Spectral approach to homogenization of hyperbolic equations with periodic coefficients, J. Diff. Equ. 264 (2018), no. 12, 7463–7522.
[4] M.A. Dorodnyi and T.A. Suslina, Homogenization of hyperbolic equations with periodic coefficients in $\mathbb{R}^d$: Sharpness of the results, St. Petersburg Math. J. 32 (2021), no. 4, 605–703.
[5] M.A. Dorodnyi and T.A. Suslina, Homogenization of hyperbolic equations with periodic coefficients: Results with correctors, in preparation.