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 CMFD, 2011, Volume 39, Pages 173–184 (Mi cmfd180)

Averaging of boundary-value problems for the Laplace operator in perforated domains with a nonlinear boundary condition of the third type on the boundary of cavities

M. N. Zubovaa, T. A. Shaposhnikovab

a Moscow
b Moscow State University, Moscow, Russia

Abstract: In this paper, the asymptotic behavior of solutions $u_\varepsilon$ of the Poisson equation in the $\varepsilon$-periodically perforated domain $\Omega_\varepsilon\subset\mathbb R^n$, $n\ge3$, with the third nonlinear boundary condition of the form $\partial_\nu u_\varepsilon+\varepsilon^{-\gamma}\sigma(x,u_\varepsilon)=\varepsilon^{-\gamma}g(x)$ on a boundary of cavities, is studied. It is supposed that the diameter of cavities has the order $\varepsilon^\alpha$ with $\alpha>1$ and any $\gamma$. Here, all types of asymptotic behavior of solutions $u_\varepsilon$, corresponding to different relations between parameters $\alpha$ and $\gamma$, are studied.

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English version:
Journal of Mathematical Sciences, 2013, 190:1, 181–193

Bibliographic databases:

UDC: 517.9

Citation: M. N. Zubova, T. A. Shaposhnikova, “Averaging of boundary-value problems for the Laplace operator in perforated domains with a nonlinear boundary condition of the third type on the boundary of cavities”, Partial differential equations, CMFD, 39, PFUR, M., 2011, 173–184; Journal of Mathematical Sciences, 190:1 (2013), 181–193

Citation in format AMSBIB
\Bibitem{ZubSha11} \by M.~N.~Zubova, T.~A.~Shaposhnikova \paper Averaging of boundary-value problems for the Laplace operator in perforated domains with a~nonlinear boundary condition of the third type on the boundary of cavities \inbook Partial differential equations \serial CMFD \yr 2011 \vol 39 \pages 173--184 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd180} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2830684} \transl \jour Journal of Mathematical Sciences \yr 2013 \vol 190 \issue 1 \pages 181--193 \crossref{https://doi.org/10.1007/s10958-013-1253-5} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84874950023} 

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This publication is cited in the following articles:
1. Ildefonso Diaz J., Gomez-Castro D., Podol'skii V A., Shaposhnikova T.A., “Characterizing the Strange Term in Critical Size Homogenization: Quasilinear Equations With a General Microscopic Boundary Condition”, Adv. Nonlinear Anal., 8:1 (2019), 679–693
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