Operator error estimates in the homogenization problem for nonstationary periodic equations

M. Sh. Birman ; T. A. Suslina

Algebra i analiz, Tome 20 (2008) no. 6, pp. 30-107.

Résumé

Matrix periodic differential operators (DO's) $\mathcal A=\mathcal A (\mathbf x,\mathbf D)$ in $L_2({\mathbb R}^d;{\mathbb C}^n)$ are considered. The operators are assumed to admit factorization of the form ${\mathcal A}={\mathcal X}^*{\mathcal X}$, where $\mathcal X$ is a homogeneous first order DO. Let ${\mathcal A}_\varepsilon={\mathcal A}(\varepsilon^{-1}{\mathbf x},{\mathbf D})$, $\varepsilon>0$. The behavior of the solutions ${\mathbf u}_\varepsilon({\mathbf x},\tau)$ of the Cauchy problem for the Schrödinger equation $i\partial_\tau{\mathbf u}_\varepsilon={\mathcal A}_\varepsilon{\mathbf u}_\varepsilon$, and also the behavior of those for the hyperbolic equation $\partial^2_\tau{\mathbf u}_\varepsilon=-{\mathcal A}_\varepsilon{\mathbf u}_\varepsilon$ is studied as $\varepsilon\to 0$. Let ${\mathbf u}_0$ be the solution of the corresponding homogenized problem. Estimates of order $\varepsilon$ are obtained for the $L_2({\mathbb R}^d;{\mathbb C}^n)$-norm of the difference ${\mathbf u}_\varepsilon-{\mathbf u}_0$ for a fixed $\tau\in{\mathbb R}$. The estimates are uniform with respect to the norm of initial data in the Sobolev space $H^s({\mathbb R}^d;{\mathbb C}^n)$, where $s=3$ in the case of the Schrödinger equation and $s=2$ in the case of the hyperbolic equation. The dependence of the constants in estimates on the time $\tau$ is traced, which makes it possible to obtain qualified error estimates for small $\varepsilon$ and large $|\tau|=O(\varepsilon^{-\alpha})$ with appropriate $\alpha1$.

Détail

Keywords: Periodic operators, nonstationary equations, Cauchy problem, threshold effect, homogenization, effective operator.

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