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

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

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
\Bibitem{Che04} \by G.~A.~Chechkin \paper Estimation of Solutions of Boundary-Value Problems in Domains with Concentrated Masses Located Periodically along the Boundary: Case of Light Masses \jour Mat. Zametki \yr 2004 \vol 76 \issue 6 \pages 928--944 \mathnet{http://mi.mathnet.ru/mz152} \crossref{https://doi.org/10.4213/mzm152} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2127504} \zmath{https://zbmath.org/?q=an:1076.35014} \transl \jour Math. Notes \yr 2004 \vol 76 \issue 6 \pages 865--879 \crossref{https://doi.org/10.1023/B:MATN.0000049687.89273.d9} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000226356700029} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-10344258632} 

• http://mi.mathnet.ru/eng/mz152
• https://doi.org/10.4213/mzm152
• http://mi.mathnet.ru/eng/mz/v76/i6/p928

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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
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
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
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