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St. Petersburg Mathematical Journal

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



Homogenization for a periodic elliptic operator in a strip with various boundary conditions

Author: N. N. Senik
Translated by: The Author
Original publication: Algebra i Analiz, tom 25 (2013), nomer 4.
Journal: St. Petersburg Math. J. 25 (2014), 647-697
MSC (2010): Primary 35B27
Published electronically: June 5, 2014
MathSciNet review: 3184621
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Abstract | References | Similar Articles | Additional Information

Abstract: A homogenization problem is considered for the periodic elliptic differential operators on $ L_2(\Pi )$, $ \Pi =\mathbb{R} \times (0, a)$, defined by the differential expression

$\displaystyle \mathcal {B}_{\lambda }^{\varepsilon } = \sum _{j=1}^2 \mathrm {D... ...repsilon , x_2)\mathrm {D}_j + \mathrm {D}_j h_j(x_1/\varepsilon , x_2) \bigr )$      
$\displaystyle + \ Q(x_1/\varepsilon , x_2) + \lambda Q_*(x_1/\varepsilon , x_2)$      

with periodic, Neumann, or Dirichlet boundary conditions. The coefficients of the expression are assumed to be periodic of period $ 1$ in the first variable and smooth in some sense in the second. Sharp-order approximations are obtained for the inverse of $ \mathcal {B}_{\lambda }^{\varepsilon }$ with respect to $ \mathbf {B}\bigl ( L_2(\Pi )\bigr )$- and $ \mathbf {B}\bigl ( L_2(\Pi ), H^1(\Pi ) \bigr )$-norms in the small $ \varepsilon $ limit with error terms of order  $ \varepsilon $.

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  • [BSu] M. Sh. Birman and T. Suslina, Periodic second-order differential operators. Threshold properties and homogenization, Algebra i Analiz 15 (2003), no. 5, 1-108; English transl., St. Petersburg Math. J. 15, 639-714. MR 2068790 (2005k:47097)
  • [Su1] T. A. Suslina, On the averaging of a periodic elliptic operator in a strip, Algebra i Analiz 16 (2004), no. 1, 269-292; English transl., St. Petersburg Math. J. 16 (2005), no. 1, 237-257. MR 2068354 (2005c:35024)
  • [Su2] -, Homogenization in the Sobolev class $ H^1(\mathbb{R}^d)$ for second-order periodic elliptic
    operators with the inclusion of first-order terms
    , Algebra i Analiz 22 (2010), no. 1, 108-222; English transl., St. Petersburg Math. J. 22 (2011), no. 1, 81-162. MR 2641084 (2011d:35041)
  • [LaU] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear equation of elliptic type, 2nd ed., Nauka, Moscow, 1973; English transl. of 1st ed., Acad. Press, New York-London, 1968. MR 0509265 (58:23009)
  • [BuCaSu] R. Bunoiu, G. Cardone, and T. Suslina, Spectral approach to homogenization of an elliptic operator periodic in some directions, Math. Methods Appl. Sci. 34 (2011), no. 9, 1075-1096. MR 2829469 (2012f:35025)
  • [S-HT] J. Sanchez-Hubert and N. Turbe, Ondes élastiques dans une bande périodique, RAIRO Modél. Math. Anal. Numér. 20 (1986), no. 3, 539-561. MR 862791 (87j:73032)

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Additional Information

N. N. Senik
Affiliation: Faculty of Physics, St. Petersburg State University, Ulyanovskaya 3, Peterhof, St. Petersburg 198504, Russia

Keywords: Homogenization, operator error estimates, periodic differential operators, effective operator, corrector
Received by editor(s): September 20, 2012
Published electronically: June 5, 2014
Additional Notes: The author was supported by RFBR (grant no. 11-01-00458-a) and by the RF Ministry of Education and Science (project 07.09.2012, no. 8501, no. 2012-1.5-12-000-1003-016)
Article copyright: © Copyright 2014 American Mathematical Society

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