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Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 2, Pages 165–194 (Mi izv2769)  

This article is cited in 1 scientific paper (total in 1 paper)

Homogenization of a mixed boundary-value problem in a domain with anisotropic fractal perforation

S. A. Nazarov, A. S. Slutskii

Institute of Problems of Mechanical Engineering, Russian Academy of Sciences

Abstract: We carry out a homogenization of a mixed boundary-value problem for a scalar elliptic equation in a rectangle with anisotropic fractal perforation, namely, the (small) size of holes is preserved in one direction, whereas it is reduced in the other when moving away from the base of the rectangle. Neumann conditions are imposed on the boundaries of the holes. A specific feature of the asymptotic constructions is the presence of several boundary layers. Explicit formulae are obtained for the homogenized differential operator and asymptotically exact error estimates are derived, and the smallness of the majorant is related to the smoothness property of the right-hand side with respect to the slow variable in the scale of Sobolev–Slobodetskii spaces.

Keywords: homogenization, anisotropic perforation, fractal structure, boundary layers.

DOI: https://doi.org/10.4213/im2769

Full text: PDF file (745 kB)
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English version:
Izvestiya: Mathematics, 2010, 74:2, 379–409

Bibliographic databases:

UDC: 517.946
MSC: Primary 35B25; Secondary 35J25, 74G70, 74Q05
Received: 11.02.2008

Citation: S. A. Nazarov, A. S. Slutskii, “Homogenization of a mixed boundary-value problem in a domain with anisotropic fractal perforation”, Izv. RAN. Ser. Mat., 74:2 (2010), 165–194; Izv. Math., 74:2 (2010), 379–409

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. V. A. Kozlov, S. A. Nazarov, “The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary”, St. Petersburg Math. J., 22:6 (2011), 941–983  mathnet  crossref  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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