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Homogenization of high order elliptic operators with periodic coefficients


Authors: A. A. Kukushkin and T. A. Suslina
Translated by: T.Suslina
Original publication: Algebra i Analiz, tom 28 (2016), nomer 1.
Journal: St. Petersburg Math. J. 28 (2017), 65-108
MSC (2010): Primary 35B27
DOI: https://doi.org/10.1090/spmj/1439
Published electronically: November 30, 2016
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Abstract: A selfadjoint strongly elliptic operator $ A_\varepsilon $ of order $ 2p$ given by the expression $ b(\mathbf {D})^* g(\mathbf {x}/\varepsilon ) b(\mathbf {D})$, $ \varepsilon >0$, is studied in $ L_2(\mathbb{R}^d;\mathbb{C}^n)$. Here $ g(\mathbf {x})$ is a bounded and positive definite $ (m\times m)$-matrix-valued function on $ \mathbb{R}^d$; it is assumed that $ g(\mathbf {x})$ is periodic with respect to some lattice. Next, $ b(\mathbf {D})=\sum _{\vert\alpha \vert=p} b_\alpha \mathbf {D}^\alpha $ is a differential operator of order $ p$ with constant coefficients; the $ b_\alpha $ are constant $ (m\times n)$-matrices. It is assumed that $ m\ge n$ and that the symbol $ b({\boldsymbol \xi })$ has maximal rank. For the resolvent $ (A_\varepsilon - \zeta I)^{-1}$ with $ \zeta \in \mathbb{C} \setminus [0,\infty )$, approximations are obtained in the norm of operators in $ L_2(\mathbb{R}^d;\mathbb{C}^n)$ and in the norm of operators acting from $ L_2(\mathbb{R}^d;\mathbb{C}^n)$ to the Sobolev space $ H^p(\mathbb{R}^d;\mathbb{C}^n)$, with error estimates depending on $ \varepsilon $ and $ \zeta $.


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Additional Information

A. A. Kukushkin
Affiliation: St.-Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia
Email: beslave@gmail.com

T. A. Suslina
Affiliation: St.-Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia
Email: t.suslina@spbu.ru

DOI: https://doi.org/10.1090/spmj/1439
Keywords: Periodic differential operators, homogenization, effective operator, corrector, operator error estimates
Received by editor(s): November 2, 2015
Published electronically: November 30, 2016
Additional Notes: Supported by RFBR (grant no. 14-01-00760) and by St. Petersburg State University (project no. 11.38.263.2014).
Article copyright: © Copyright 2016 American Mathematical Society