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Sbornik: Mathematics, 2021, Volume 212, Issue 8, Pages 1068–1121
DOI: https://doi.org/10.1070/SM9435
(Mi sm9435)
 

This article is cited in 17 scientific papers (total in 17 papers)

Uniform convergence and asymptotics for problems in domains finely perforated along a prescribed manifold in the case of the homogenized Dirichlet condition

D. I. Borisovabc, A. I. Mukhametrakhimovad

a Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, Russia
b Bashkir State University, Ufa, Russia
c University of Hradec Králové, Hradec Králové, Czech Republic
d Bashkir State Pedagogical University n. a. M. Akmulla, Ufa, Russia
References:
Abstract: A boundary value problem for a second-order elliptic equation with variable coefficients is considered in a multidimensional domain which is perforated by small holes along a prescribed manifold. Minimal natural conditions are imposed on the holes. In particular, all of these are assumed to be of approximately the same size and have a prescribed minimal distance to neighbouring holes, which is also a small parameter. The shape of the holes and their distribution along the manifold are arbitrary. The holes are divided between two sets in an arbitrary way. The Dirichlet condition is imposed on the boundaries of holes in the first set and a nonlinear Robin boundary condition is imposed on the boundaries of holes in the second. The sizes and distribution of holes with the Dirichlet condition satisfy a simple and easily verifiable condition which ensures that these holes disappear after homogenization and a Dirichlet condition on the manifold in question arises instead. We prove that the solution of the perturbed problem converges to the solution of the homogenized one in the $W_2^1$-norm uniformly with respect to the right-hand side of the equation, and an estimate for the rate of convergence that is sharp in order is deduced. The full asymptotic solution of the perturbed problem is also constructed in the case when the holes form a periodic set arranged along a prescribed hyperplane.
Bibliography: 32 titles.
Keywords: perforated domain, boundary value problem, homogenization, uniform convergence, estimate for the rate of convergence, asymptotic.
Funding agency Grant number
Russian Science Foundation 20-11-19995
The research presented in § 5 was carried out using funding from the Russian Science Foundation under grant no. 20-11-19995.
Received: 30.04.2020 and 28.10.2020
Russian version:
Matematicheskii Sbornik, 2021, Volume 212, Number 8, Pages 33–88
DOI: https://doi.org/10.4213/sm9435
Bibliographic databases:
Document Type: Article
UDC: 517.956+517.958
MSC: 35J15, 35B27
Language: English
Original paper language: Russian
Citation: D. I. Borisov, A. I. Mukhametrakhimova, “Uniform convergence and asymptotics for problems in domains finely perforated along a prescribed manifold in the case of the homogenized Dirichlet condition”, Mat. Sb., 212:8 (2021), 33–88; Sb. Math., 212:8 (2021), 1068–1121
Citation in format AMSBIB
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  • This publication is cited in the following 17 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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