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.
Citation:
S. A. Nazarov, A. S. Slutskii, “Homogenization of a mixed boundary-value problem in a domain with anisotropic fractal perforation”, Izv. Math., 74:2 (2010), 379–409
\Bibitem{NazSlu10}
\by S.~A.~Nazarov, A.~S.~Slutskii
\paper Homogenization of a~mixed boundary-value problem in a~domain with anisotropic fractal perforation
\jour Izv. Math.
\yr 2010
\vol 74
\issue 2
\pages 379--409
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Linking options:
https://www.mathnet.ru/eng/im2769
https://doi.org/10.1070/IM2010v074n02ABEH002490
https://www.mathnet.ru/eng/im/v74/i2/p165
This publication is cited in the following 2 articles:
D.I. Borisov, “Homogenization for operators with arbitrary perturbations in coefficients”, Journal of Differential Equations, 369 (2023), 41
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
Citation in format AMSBIB
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