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 Izv. RAN. Ser. Mat., 1996, Volume 60, Issue 1, Pages 133–164 (Mi izv65)

$G$-compactness of sequences of non-linear operators of Dirichlet problems with a variable domain of definition

A. A. Kovalevsky

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

Abstract: For a sequence of operators $A_s\colon\overset{\circ}{W} ^{1,m}(\Omega_s)\to(\overset{\circ}{W} ^{1,m}(\Omega_s))^*$ in divergence form we prove a theorem concerning the choice of a subsequence that $G$-converges to the operator $\widehat A\colon\overset{\circ}{W} ^{1,m}(\Omega)\to(\overset{\circ}{W} ^{1,m}(\Omega))^*$ with the same leading coefficients as the operator $A_s$ and some additional lower coefficient $b(x,u)$. We give a procedure for constructing the function $b(x,u)$. We discuss the question of whether the principal condition under which the choice theorem is established is necessary. We prove criteria for this condition to hold.

DOI: https://doi.org/10.4213/im65

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English version:
Izvestiya: Mathematics, 1996, 60:1, 137–168

Bibliographic databases:

MSC: Primary 35J65, 49L10, 49L15; Secondary 47H15, 47H17, 35B99, 35D99

Citation: A. A. Kovalevsky, “$G$-compactness of sequences of non-linear operators of Dirichlet problems with a variable domain of definition”, Izv. RAN. Ser. Mat., 60:1 (1996), 133–164; Izv. Math., 60:1 (1996), 137–168

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv65
• https://doi.org/10.4213/im65
• http://mi.mathnet.ru/eng/izv/v60/i1/p133

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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. 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), 1151–1156
2. Alexander Kovalevsky, Francesco Nicolosi, “Integral estimates for solutions of some degenerate local variational inequalities”, Applicable Analysis, 73:3-4 (1999), 425
3. Kovalevskii A.A., “A necessary condition for the strong G-convergence of nonlinear operators of Dirichlet problems with variable domain”, Differential Equations, 36:4 (2000), 599–604
4. Afonina N.E., Gromov V.G., Kovalev V.L., “Investigation of the Influence of Different Heterogeneous Recombination Mechanisms on the Heat Fluxes to a Catalytic Surface in Dissociated Carbon Dioxide”, Fluid Dynamics, 37:1 (2002), 117–125
5. Kovalev V.L., Afonina N.E., Gromov V.G., “Effect of different geterogeneous recombination mechanisms on heat fluxes to catalytic surfaces in carbon dioxide”, 4Th European Symposium on Aerothermodynamics for Space Vehicles, Proceedings, ESA Special Publications, 487, 2002, 131–136
6. Kovalev V., Afonina N., Gromov V., “Effect of physical adsorption on heat fluxes to catalytic surfaces in carbon dioxide”, Rarefied Gas Dynamics, AIP Conference Proceedings, 663, 2003, 1001–1007
7. Mel'nyk T.A., Sivak O.A., “Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains”, Asymptot Anal, 75:1–2 (2011), 79–92
8. Alexander A. Kovalevsky, “On -functions with a very singular behaviour”, Nonlinear Analysis: Theory, Methods & Applications, 85 (2013), 66
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