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Operator error estimates for homogenization of the elliptic Dirichlet problem in a bounded domain


Authors: M. A. Pakhnin and T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 24 (2012), nomer 6.
Journal: St. Petersburg Math. J. 24 (2013), 949-976
MSC (2010): Primary 35B27
Published electronically: September 23, 2013
MathSciNet review: 3097556
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Abstract: Let $ \mathcal {O} \subset \mathbb{R}^d$ be a bounded domain of class $ C^{1,1}$. In the Hilbert space $ L_2(\mathcal {O};\mathbb{C}^n)$, a matrix elliptic second order differential operator $ \mathcal {A}_{D,\varepsilon }$ is considered with the Dirichlet boundary condition. Here $ \varepsilon >0$ is a small parameter. The coefficients of the operator are periodic and depend on $ \mathbf {x}/\varepsilon $. Approximation is found for the operator $ \mathcal {A}_{D,\varepsilon }^{-1}$ in the norm of operators acting from $ L_2(\mathcal {O};\mathbb{C}^n)$ to the Sobolev space $ H^1(\mathcal {O};\mathbb{C}^n)$ with an error term of $ O(\sqrt {\varepsilon })$. This approximation is given by the sum of the operator $ (\mathcal {A}^0_D)^{-1}$ and the first order corrector, where $ \mathcal {A}^0_D$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.


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Additional Information

M. A. Pakhnin
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: mpakhnin@yandex.ru

T. A. Suslina
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: suslina@list.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01274-X
Keywords: Periodic differential operators, homogenization, effective operator, operator error estimates
Received by editor(s): July 2, 2012
Published electronically: September 23, 2013
Additional Notes: Supported by RFBR (grant no. 11-01-00458-a) and the Program of support for leading scientific schools (grant NSh-357.2012.1).
Article copyright: © Copyright 2013 American Mathematical Society