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Homogenization of elliptic systems with periodic coefficients: operator error estimates in $ L_2(\mathbb{R}^d)$ with corrector taken into account

Author: T. A. Suslina
Translated by: the author
Original publication: Algebra i Analiz, tom 26 (2014), nomer 4.
Journal: St. Petersburg Math. J. 26 (2015), 643-693
MSC (2010): Primary 35B27
Published electronically: May 6, 2015
MathSciNet review: 3289189
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Abstract: A matrix elliptic selfadjoint second order differential operator (DO) $ {\mathcal B}_{\varepsilon }$ with rapidly oscillating coefficients is considered in $ L_2(\mathbb{R}^d;\mathbb{C}^n)$. The principal part $ b(\mathbf {D})^* g(\varepsilon ^{-1}\mathbf {x})b(\mathbf {D})$ of this operator is given in a factorized form, where $ g$ is a periodic, bounded, and positive definite matrix-valued function and $ b(\mathbf {D})$ is a matrix first order DO whose symbol is a matrix of maximal rank. The operator $ {\mathcal B}_\varepsilon $ also includes first and zero order terms with unbounded coefficients. The problem of homogenization in the small period limit is studied. For the generalized resolvent of $ \mathcal {B}_\varepsilon $, approximation in the $ L_2(\mathbb{R}^d;\mathbb{C}^n)$-operator norm with an error $ O(\varepsilon ^2)$ is obtained. The principal term of this approximation is given by the generalized resolvent of the effective operator $ {\mathcal B}^0$ with constant coefficients. The first order corrector is taken into account. The error estimate obtained is order sharp; the constants in estimates are controlled in terms of the problem data. General results are applied to homogenization problems for the Schrödinger operator and the two-dimensional Pauli operator with singular rapidly oscillating potentials.

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Additional Information

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

Keywords: Homogenization, effective operator, corrector, operator error estimates
Received by editor(s): January 12, 2014
Published electronically: May 6, 2015
Additional Notes: Supported by RFBR (grant no.14-01-00760) and by St.Petersburg State University (grant no.
Article copyright: © Copyright 2015 American Mathematical Society

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