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Homogenization of elliptic operators with periodic coefficients in dependence of the spectral parameter

Author: T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 27 (2015), nomer 4.
Journal: St. Petersburg Math. J. 27 (2016), 651-708
MSC (2010): Primary 35B27
Published electronically: June 2, 2016
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Abstract: Differential expressions of the form $ b(\mathbf {D})^* g(\mathbf {x}/\varepsilon ) b(\mathbf {D})$, $ \varepsilon >0$, are considered, where a matrix-valued function $ g(\mathbf {x})$ in $ \mathbb{R}^d$ is assumed to be bounded, positive definite, and periodic with respect to some lattice; $ b(\mathbf {D})=\sum _{l=1}^d b_l D_l$ is a first order differential operator with constant coefficients. The symbol $ b({\boldsymbol \xi })$ is subject to some condition ensuring strong ellipticity. The operator in $ L_2(\mathbb{R}^d;\mathbb{C}^n)$ given by the expression $ b(\mathbf {D})^* g(\mathbf {x}/\varepsilon ) b(\mathbf {D})$ is denoted by $ \mathcal {A}_\varepsilon $. Let $ \mathcal {O} \subset \mathbb{R}^d$ be a bounded domain of class $ C^{1,1}$. The operators $ \mathcal {A}_{D,\varepsilon }$ and $ \mathcal {A}_{N,\varepsilon }$ under study are generated in the space $ L_2(\mathcal {O};\mathbb{C}^n)$ by the above expression with the Dirichlet or Neumann boundary conditions. Approximations in various operator norms for the resolvents $ (\mathcal {A}_\varepsilon - \zeta I)^{-1}$, $ (\mathcal {A}_{D,\varepsilon }- \zeta I)^{-1}$, $ (\mathcal {A}_{N,\varepsilon }-\zeta I)^{-1}$ are obtained with error estimates depending on $ \varepsilon $ and $ \zeta $.

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Additional Information

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

Keywords: Periodic differential operators, Dirichlet problem, Neumann problem, homogenization, effective operator, corrector, operator error estimates
Received by editor(s): December 10, 2014
Published electronically: June 2, 2016
Additional Notes: Supported by RFBR (grant no. 14-01-00760a) and by St. Petersburg State University (grant no.
Article copyright: © Copyright 2016 American Mathematical Society