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St. Petersburg Mathematical Journal

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

 
 

 

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
MathSciNet review: 3591067
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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, $b(\mathbf {D})=\sum _{|\alpha |=p} b_\alpha \mathbf {D}^\alpha$ 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

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