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 Izv. Akad. Nauk SSSR Ser. Mat., 1990, Volume 54, Issue 5, Pages 990–1020 (Mi izv1059)

Asymptotic solution of a variational inequality modelling a friction

S. A. Nazarov

Abstract: The problem of minimizing the nondifferentiable functional
$$\mu^2(\nabla u,\nabla u)_\Omega\times (u,u)_\Omega -2(f,u)_\Omega+\gamma(|u|,g)_{\partial\Omega}$$
is considered. An asymptotic solution of the corresponding variational inequality is constructed and justified under the assumption that $\mu$ or $\gamma$ is a small parameter. Also, formal asymptotic representations are obtained for singular surfaces which characterize a change in the type of boundary conditions. For $\mu\to 0$ a modification of the Vishik–Lyusternik method is used, and exponential boundary layers arise. If $\gamma\to 0$, then the boundary layer has only power growth; the principal term of the asymptotic expansion of the solution of the problem in a multidimensional region $\Omega$ and the complete asymptotic expansion for the case $\Omega\subset\mathbf R^2$ are obtained.

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English version:
Mathematics of the USSR-Izvestiya, 1991, 37:2, 337–369

Bibliographic databases:

UDC: 517.946
MSC: Primary 35C20; Secondary 35B25, 35J25, 49A29

Citation: S. A. Nazarov, “Asymptotic solution of a variational inequality modelling a friction”, Izv. Akad. Nauk SSSR Ser. Mat., 54:5 (1990), 990–1020; Math. USSR-Izv., 37:2 (1991), 337–369

Citation in format AMSBIB
\Bibitem{Naz90} \by S.~A.~Nazarov \paper Asymptotic solution of a variational inequality modelling a friction \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1990 \vol 54 \issue 5 \pages 990--1020 \mathnet{http://mi.mathnet.ru/izv1059} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1086083} \zmath{https://zbmath.org/?q=an:0733.49015|0713.49012} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1991IzMat..37..337N} \transl \jour Math. USSR-Izv. \yr 1991 \vol 37 \issue 2 \pages 337--369 \crossref{https://doi.org/10.1070/IM1991v037n02ABEH002067} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. I. I. Argatov, S. A. Nazarov, “Asymptotic solution of the Signorini problem with an obstacle on a thin elongated set”, Sb. Math., 187:10 (1996), 1411–1442
2. O. V. Izotova, S. A. Nazarov, “An asymptotic solution to the Signorini problem about a beam laying on two rigid bases”, J. Math. Sci. (N. Y.), 138:2 (2006), 5503–5513
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