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 Mat. Sb. (N.S.), 1977, Volume 103(145), Number 4(8), Pages 614–629 (Mi msb2932)

This article is cited in 16 scientific papers (total in 16 papers)

The first boundary value problem in domains with a complicated boundary for higher order equations

E. Ya. Khruslov

Abstract: The first boundary value problem is considered for an elliptic selfadjoint operator $L$ of order $2m$ in a domain $\Omega^{(s)}$ of complicated structure of the form $\Omega^{(s)}=\Omega\setminus F^{(s)}$, where $\Omega$ is a comparatively simple domain in $\mathbf R_n$ ($n\geqslant2$) and $F^{(s)}$ is a closed, connected, highly fragmented set in $\Omega$. The asymptotic behavior of the resolvent $R^{(s)}$ of this problem is studied for $s\to\infty$ when the set $F^{(s)}$ becomes ever more fragmented and is disposed volumewise in $\Omega$ so that the distance from $F^{(s)}$ to any point $x\in\Omega$ tends to zero.
It is shown that $R^{(s)}$ converges in norm to the resolvent $R^c$ of an operator $L+c(x)$, which is considered in the simple domain $\Omega$ under null conditions in $\partial\Omega$. A massivity characteristic of the sets $F^{(s)}$ (of capacity type) is introduced, which is used to formulate necessary and sufficient conditions for convergence, and the function $c(x)$ is described.
Bibliography: 7 titles.

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English version:
Mathematics of the USSR-Sbornik, 1977, 32:4, 535–549

Bibliographic databases:

UDC: 517.946
MSC: Primary 35J40; Secondary 47B25
Received: 09.11.1976

Citation: E. Ya. Khruslov, “The first boundary value problem in domains with a complicated boundary for higher order equations”, Mat. Sb. (N.S.), 103(145):4(8) (1977), 614–629; Math. USSR-Sb., 32:4 (1977), 535–549

Citation in format AMSBIB
\Bibitem{Khr77} \by E.~Ya.~Khruslov \paper The first boundary value problem in~domains with a~complicated boundary for higher order equations \jour Mat. Sb. (N.S.) \yr 1977 \vol 103(145) \issue 4(8) \pages 614--629 \mathnet{http://mi.mathnet.ru/msb2932} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=463679} \zmath{https://zbmath.org/?q=an:0359.35023} \transl \jour Math. USSR-Sb. \yr 1977 \vol 32 \issue 4 \pages 535--549 \crossref{https://doi.org/10.1070/SM1977v032n04ABEH002405} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1977GL81400009} 

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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. E. Ya. Khruslov, “The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain”, Math. USSR-Sb., 35:2 (1979), 266–282
2. S. M. Kozlov, “Averaging of random operators”, Math. USSR-Sb., 37:2 (1980), 167–180
3. L. V. Berlyand, “The asymptotic behaviour of solutions of the first boundary-value problem of elasticity theory in domains with a finely-grained boundary”, Russian Math. Surveys, 38:6 (1983), 11–112
4. G. Maso, G. Paderni, “Variational inequalities for the biharmonic operator with variable obstacles”, Annali di Matematica, 153:1 (1988), 203
5. Michele Balzano, “Random relaxed Dirichlet problems”, Annali di Matematica pura ed applicata, 153:1 (1988), 133
6. Satoshi Kaizu, “Behavior of solutions of the Poisson equation under fragmentation of the boundary of the domain”, Japan J Appl Math, 7:1 (1990), 77
7. Satoshi Kaizu, “The Poisson Equation with Nonautonomous Semilinear Boundary Conditions in Domains with Many Time Holes”, SIAM J Math Anal, 22:5 (1991), 1222
8. A. A. Kovalevsky, “$G$-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain”, Russian Acad. Sci. Izv. Math., 44:3 (1995), 431–460
9. A. A. Kovalevsky, “$G$-compactness of sequences of non-linear operators of Dirichlet problems with a variable domain of definition”, Izv. Math., 60:1 (1996), 137–168
10. dalMaso, G, “Asymptotic behavior of solutions of Dirichlet problems”, Bollettino Della Unione Matematica Italiana, 11A:2 (1997), 253
11. Gianni Dal Maso, Igor V. Skrypnik, “Asymptotic behaviour of nonlinear Dirichlet problems in perforated domains”, Annali di Matematica, 174:1 (1998), 13
12. D. Maso, I. V. Skrypnik, “Asymptotic behaviour of nonlinear elliptic higher order equations in perforated domains”, J Anal Math, 79:1 (1999), 63
13. Kovalevsky, A, “An effect of double homogenization for Dirichlet problems in variable domains of general structure”, Comptes Rendus de l Academie Des Sciences Serie i-Mathematique, 328:12 (1999), 1151
14. Alexander Kovalevsky, Francesco Nicolosi, “Integral estimates for solutions of some degenerate local variational inequalities”, Applicable Analysis, 73:3-4 (1999), 425
15. Kovalevskii, AA, “A necessary condition for the strong G-convergence of nonlinear operators of Dirichlet problems with variable domain”, Differential Equations, 36:4 (2000), 599
16. S. D’Asero, D.V. Larin, “Degenerate nonlinear higher-order elliptic problems in domains with fine-grained boundary”, Nonlinear Analysis: Theory, Methods & Applications, 64:4 (2006), 788
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