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 Algebra i Analiz, 2009, Volume 21, Issue 4, Pages 126–173 (Mi aa1147)

Research Papers

Homogenization of the mixed boundary value problem for a formally self-adjoint system in a periodically perforated domain

G. Cardonea, A. Corbo Espositob, S. A. Nazarovc

a University of Sannio, Department of Engineering, Benevento, Italy
b University of Cassino, Department of Automation, Electromagnetism Information and Industrial Mathematics, Cassino, Italy
c Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia

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 $O(h^\delta)$ $O(h^\delta)$ of homogenization depends on the smoothness properties of the problem data.

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English version:
St. Petersburg Mathematical Journal, 2010, 21:4, 601–634

Bibliographic databases:

MSC: 35J57

Citation: G. Cardone, A. Corbo Esposito, S. A. Nazarov, “Homogenization of the mixed boundary value problem for a formally self-adjoint system in a periodically perforated domain”, Algebra i Analiz, 21:4 (2009), 126–173; St. Petersburg Math. J., 21:4 (2010), 601–634

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Nazarov S.A., Thäter G., “The Stokes problem in a periodic layer”, Math. Nachr., 284:10 (2011), 1201–1218
2. Cardone G., Nazarov S.A., Piatnitski A.L., “On the Rate of Convergence for Perforated Plates with a Small Interior Dirichlet Zone”, Z. Angew. Math. Phys., 62:3 (2011), 439–468
3. M. M. Karchevskii, R. R. Shagidullin, “O kraevykh zadachakh dlya ellipticheskikh sistem uravnenii vtorogo poryadka divergentnogo vida”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 157, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2015, 93–103
4. Borisov D. Cardone G. Durante T., “Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve”, Proc. R. Soc. Edinb. Sect. A-Math., 146:6 (2016), 1115–1158
5. S. A. Nazarov, “Homogenization of Kirchhoff plates with oscillating edges and point supports”, Izv. Math., 84:4 (2020), 722–779
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