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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

Operator error estimates in the homogenization problem for nonstationary periodic equations

Authors: M. Sh. Birman and T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 20 (2008), nomer 6.
Journal: St. Petersburg Math. J. 20 (2009), 873-928
MSC (2000): Primary 35B27
Published electronically: October 1, 2009
MathSciNet review: 2530894
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Abstract: 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 a 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 $ \vert\tau\vert =O(\varepsilon^{-\alpha})$ with appropriate $ \alpha<1$.

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Additional Information

M. Sh. Birman
Affiliation: Department of Physics, St. Petersburg State University, Ul’yanovskaya 3, Petrodvorets, 198504 St. Petersburg, Russia

T. A. Suslina
Affiliation: Department of Physics, St. Petersburg State University, Ul’yanovskaya 3, Petrodvorets, 198504 St. Petersburg, Russia

PII: S 1061-0022(09)01077-2
Keywords: Periodic operators, nonstationary equations, Cauchy problem, threshold effect, homogenization, effective operator
Received by editor(s): August 10, 2008
Published electronically: October 1, 2009
Additional Notes: Supported by RFBR (grant no. 08-01-00209-a) and “Scientific Schools” grant no. 816.2008.1.
Dedicated: To the memory of Tatyana Petrovna Il′ina
Article copyright: © Copyright 2009 American Mathematical Society