Homogenization for a periodic elliptic operator in a strip with various boundary conditions
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N. N. Senik
Translated by: The Author - St. Petersburg Math. J. 25 (2014), 647-697
- DOI: https://doi.org/10.1090/S1061-0022-2014-01311-8
- Published electronically: June 5, 2014
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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 \begin{align*} \mathcal {B}_{\lambda }^{\varepsilon } = \sum _{j=1}^2 \mathrm {D}_j g_j(x_1/\varepsilon , x_2)\mathrm {D}_j + \sum _{j=1}^2 \bigl ( h_{j}^{*}(x_1/\varepsilon , x_2)\mathrm {D}_j + \mathrm {D}_j h_j(x_1/\varepsilon , x_2) \bigr )& \\ + \ Q(x_1/\varepsilon , x_2) + \lambda Q_*(x_1/\varepsilon , x_2)& \end{align*} 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 $\varepsilon$.References
- M. Sh. Birman and T. A. Suslina, Periodic second-order differential operators. Threshold properties and averaging, Algebra i Analiz 15 (2003), no. 5, 1–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639–714. MR 2068790, DOI 10.1090/S1061-0022-04-00827-1
- T. A. Suslina, On the averaging of a periodic elliptic operator in a strip, Algebra i Analiz 16 (2004), no. 1, 269–292 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 1, 237–257. MR 2068354, DOI 10.1090/S1061-0022-04-00849-0
- T. A. Suslina, Homogenization in the Sobolev class $H^1(\Bbb R^d)$ for second-order periodic elliptic operators with the inclusion of first-order terms, Algebra i Analiz 22 (2010), no. 1, 108–222 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 22 (2011), no. 1, 81–162. MR 2641084, DOI 10.1090/S1061-0022-2010-01135-X
- O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
- 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, DOI 10.1002/mma.1424
- Jacqueline Sanchez-Hubert and Nicole Turbé, Ondes élastiques dans une bande périodique, RAIRO Modél. Math. Anal. Numér. 20 (1986), no. 3, 539–561 (French, with English summary). MR 862791, DOI 10.1051/m2an/1986200305391
Bibliographic Information
- N. N. Senik
- Affiliation: Faculty of Physics, St. Petersburg State University, Ulyanovskaya 3, Peterhof, St. Petersburg 198504, Russia
- Email: N.N.Senik@gmail.com
- 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)
- © Copyright 2014 American Mathematical Society
- Journal: St. Petersburg Math. J. 25 (2014), 647-697
- MSC (2010): Primary ~35B27
- DOI: https://doi.org/10.1090/S1061-0022-2014-01311-8
- MathSciNet review: 3184621