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 Izv. RAN. Ser. Mat., 1994, Volume 58, Issue 3, Pages 3–35 (Mi izv784)

$G$-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain

A. A. Kovalevsky

Institute of Applied Mathematics and Mechanics, Ukraine National Academy of Sciences

Abstract: The concepts of $G$-convergence and strong $G$-convergence of a sequence of elliptic operators $A_s\colon W^{1,m}(\Omega_s)\to(W^{1,m}(\Omega_s))^*$ are studied, where $\Omega_s$, $s=1,2,…$, are perforated domains contained in a bounded domain $\Omega\subset\mathbf R^n$. It is established that $G$-convergence of the operators $A_s$ is accompanied by convergence of solutions of certain equations and variational inequalities connected with the operators $A_s$ and a theorem on selection from the sequence $\{A_s\}$ of a strongly $G$-convergent subsequence. It is shown that under the condition of periodicity of the perforation of domains $\Omega_s$ and certain assumptions regarding the coefficients of the operators $A_s$, strong $G$-convergence of $\{A_s\}$ to an operator $A\colon W^{1,m}(\Omega)\to(W^{1,m}(\Omega))^*$ holds with effectively computable coefficients.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1995, 44:3, 431–460

Bibliographic databases:

UDC: 517.98
MSC: 35J60, 35J85, 47F05

Citation: A. A. Kovalevsky, “$G$-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain”, Izv. RAN. Ser. Mat., 58:3 (1994), 3–35; Russian Acad. Sci. Izv. Math., 44:3 (1995), 431–460

Citation in format AMSBIB
\Bibitem{Kov94} \by A.~A.~Kovalevsky \paper $G$-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain \jour Izv. RAN. Ser. Mat. \yr 1994 \vol 58 \issue 3 \pages 3--35 \mathnet{http://mi.mathnet.ru/izv784} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1286837} \zmath{https://zbmath.org/?q=an:0836.35014} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1995IzMat..44..431K} \transl \jour Russian Acad. Sci. Izv. Math. \yr 1995 \vol 44 \issue 3 \pages 431--460 \crossref{https://doi.org/10.1070/IM1995v044n03ABEH001607} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RQ68000001} 

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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. P. I. Kogut, “G*-Convergence of Nonlinear Operators in the Theory of Homogenization of Control Objects”, Cybern Syst Anal, 41:5 (2005), 670
2. Alexander A. Kovalevsky, “Obstacle problems in variable domains”, Complex Variables and Elliptic Equations, 2011, 1
3. A. A. Kovalevsky, “On the convergence of solutions of variational problems with bilateral obstacles in variable domains”, Proc. Steklov Inst. Math. (Suppl.), 296, suppl. 1 (2017), 151–163
4. Kovalevsky A.A., “On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains”, Nonlinear Anal.-Theory Methods Appl., 147 (2016), 63–79
5. A. A. Kovalevskii, “Variatsionnye zadachi s odnostoronnimi potochechno funktsionalnymi ogranicheniyami v peremennykh oblastyakh”, Tr. IMM UrO RAN, 23, no. 2, 2017, 133–150
6. Alexander A. Kovalevsky, “Convergence of solutions of bilateral problems in variable domains and related questions”, Ural Math. J., 3:2 (2017), 51–66
7. A. A. Kovalevskii, “O skhodimosti reshenii variatsionnykh zadach s neyavnymi ogranicheniyami, zadannymi bystro ostsilliruyuschimi funktsiyami”, Tr. IMM UrO RAN, 24, no. 2, 2018, 107–122
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