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Homogenization of the mixed boundary-value problem for a formally selfadjoint elliptic system in a periodically punched domain


Authors: G. Cardone, A. Corbo Esposito and S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 21 (2009), nomer 4.
Journal: St. Petersburg Math. J. 21 (2010), 601-634
MSC (2010): Primary 35J57
DOI: https://doi.org/10.1090/S1061-0022-2010-01108-7
Published electronically: May 20, 2010
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Abstract: A generalized Gårding-Korn inequality is established in a domain $ \Omega(h)\subset{\mathbb{R}}^n$ with a small, of size $ O(h)$, periodic perforation, without any restrictions on the shape of the periodicity cell, except for the usual assumptions that the boundary is Lipschitzian, which ensures the Korn inequality in a general domain. Homogenization is performed for a formally selfadjoint elliptic system of second order differential equations with the Dirichlet or Neumann conditions on the outer or inner parts of the boundary, respectively; the data of the problem are assumed to satisfy assumptions of two types: additional smoothness is required from the dependence on either the ``slow'' variables $ x$, or the ``fast'' variables $ y=h^{-1}x$. It is checked that the exponent $ \delta\in(0,1/2]$ in the accuracy $ O(h^\delta)$ of homogenization depends on the smoothness properties of the problem data.


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Additional Information

G. Cardone
Affiliation: Department of Engineering, University of Sannio, Corso Garibaldi, 107, 84100 Benevento, Italy
Email: giuseppe.cardone@unisannio.it

A. Corbo Esposito
Affiliation: Department of Automation, Electromagnetism, Information and Industrial Mathematics, University of Cassino, Via G. Di Biasio, 43, 03043 Cassino (FR), Italy
Email: corbo@unicas.it

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol′shoi Prospekt V.O. 61, St. Petersburg 199178, Russia
Email: srgnazarov@yahoo.co.uk

DOI: https://doi.org/10.1090/S1061-0022-2010-01108-7
Keywords: G{\aa }rding--Korn inequality, homogenization, formally selfadjoint elliptic system, rate of convergence
Received by editor(s): November 24, 2008
Published electronically: May 20, 2010
Additional Notes: S. A. Nazarov was supported by RFBR (grant no. 09-00759).
Article copyright: © Copyright 2010 American Mathematical Society

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