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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients

Author: T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 29 (2017), nomer 2.
Journal: St. Petersburg Math. J. 29 (2018), 325-362
MSC (2010): Primary 35B27
Published electronically: March 12, 2018
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Abstract: Let $ {\mathcal O} \subset \mathbb{R}^d$ be a bounded domain of class $ C^{2p}$. The object under study is a selfadjoint strongly elliptic operator $ A_{D,\varepsilon }$ of order $ 2p$, $ p\geq 2$, in $ L_2({\mathcal O};\mathbb{C}^n)$, given by the expression $ b(\mathbf D)^* g(\mathbf x/\varepsilon ) b(\mathbf D)$, $ \varepsilon >0$, with the Dirichlet boundary conditions. Here $ g(\mathbf x)$ is a bounded and positive definite $ (m\times m)$-matrix-valued function in $ \mathbb{R}^d$, periodic with respect to some lattice; $ b(\mathbb{D})=\sum _{\vert\alpha \vert=p} b_\alpha \mathbf {D}^\alpha $ is a differential operator of order $ p$ with constant coefficients; and the $ b_\alpha $ are constant $ (m\times n)$-matrices. It is assumed that $ m\geq n$ and the symbol $ b({\boldsymbol \xi })$ has maximal rank. Approximations are found for the resolvent $ (A_{D,\varepsilon } - \zeta I)^{-1}$ in the $ L_2({\mathcal O};\mathbb{C}^n)$-operator norm and in the norm of operators acting from $ L_2({\mathcal O};\mathbb{C}^n)$ to $ H^p({\mathcal O};\mathbb{C}^n)$, with error estimates depending on $ \varepsilon $ and $ \zeta $.

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Additional Information

T. A. Suslina
Affiliation: St. Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia

Keywords: Periodic differential operators, higher-order elliptic equations, Dirichlet problem, homogenization, effective operator, corrector, operator error estimates
Received by editor(s): August 24, 2016
Published electronically: March 12, 2018
Additional Notes: Supported by RFBR (project no. 16-01-00087)
Dedicated: To the memory of Vladimir Savel’evich Buslaev
Article copyright: © Copyright 2018 American Mathematical Society

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