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Mat. Zametki, 2004, Volume 76, Issue 6, Pages 928–944 (Mi mz152)  

This article is cited in 3 scientific papers (total in 3 papers)

Estimation of Solutions of Boundary-Value Problems in Domains with Concentrated Masses Located Periodically along the Boundary: Case of Light Masses

G. A. Chechkin

M. V. Lomonosov Moscow State University

Abstract: We study the asymptotic behavior of solutions and eigenelements of boundary-value problems with rapidly alternating type of boundary conditions in the domain $\Omega\subset\mathbb R^n$. The density, which depends on a small parameter $\varepsilon$, is of the order of $O(1)$ outside small inclusions, where the density is of the order of $O((\varepsilon \delta)^{-m})$. These domains, i.e., concentrated masses of diameter $O(\varepsilon \delta)$, are located near the boundary at distances of the order of $O(\delta)$ from each other, where $\delta=\delta(\varepsilon )\to0$. We pose the Dirichlet condition (respectively, the Neumann condition) on the parts of the boundary $\partial\Omega$ that are tangent (respectively, lying outside) the concentrated masses. We estimate the deviations of the solutions of the limit (averaged) problems from the solutions of the original problems in the norm of the Sobolev space $W_2^1$ for $m<2$.

DOI: https://doi.org/10.4213/mzm152

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English version:
Mathematical Notes, 2004, 76:6, 865–879

Bibliographic databases:

UDC: 517.956.226
Received: 27.02.2003

Citation: G. A. Chechkin, “Estimation of Solutions of Boundary-Value Problems in Domains with Concentrated Masses Located Periodically along the Boundary: Case of Light Masses”, Mat. Zametki, 76:6 (2004), 928–944; Math. Notes, 76:6 (2004), 865–879

Citation in format AMSBIB
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\pages 928--944
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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. G. A. Chechkin, “Asymptotic expansions of eigenvalues and eigenfunctions of an elliptic operator in a domain with many “light” concentrated masses situated on the boundary. Two-dimensional case”, Izv. Math., 69:4 (2005), 805–846  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Chechkin G.A., Koroleva Yu.O., Persson L.-E., “On the precise asymptotics of the constant in Friedrich's inequality for functions vanishing on the part of the boundary with microinhomogeneous structure”, Journal of Inequalities and Applications, 2007, 34138  mathscinet  zmath  isi
    3. Chechkin G.A., Cioranescu D., Damlamian A., Piatnitski A.L., “On Boundary Value Problem with Singular Inhomogeneity Concentrated on the Boundary”, J. Math. Pures Appl., 98:2 (2012), 115–138  crossref  mathscinet  zmath  isi  elib  scopus  scopus
  • Математические заметки Mathematical Notes
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