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

This article is cited in 7 scientific papers (total in 7 papers)

$G$-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
Received: 25.12.1992

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
\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
\jour Russian Acad. Sci. Izv. Math.
\yr 1995
\vol 44
\issue 3
\pages 431--460

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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  crossref  mathscinet  zmath
    2. Alexander A. Kovalevsky, “Obstacle problems in variable domains”, Complex Variables and Elliptic Equations, 2011, 1  crossref
    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  mathnet  crossref  mathscinet  isi  elib
    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  crossref  mathscinet  zmath  isi  scopus
    5. A. A. Kovalevskii, “Variatsionnye zadachi s odnostoronnimi potochechno funktsionalnymi ogranicheniyami v peremennykh oblastyakh”, Tr. IMM UrO RAN, 23, no. 2, 2017, 133–150  mathnet  crossref  elib
    6. Alexander A. Kovalevsky, “Convergence of solutions of bilateral problems in variable domains and related questions”, Ural Math. J., 3:2 (2017), 51–66  mathnet  crossref
    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  mathnet  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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