Abstract
A concept ofG-convergence of operatorsA s:W s → W *s to an operatorA:W →W * is introduced and studied under a certain relationship between Banach spacesW s,s=1,2, ..., and a Banach spaceW. It is shown that conditions establishing this relationship for abstract spaces are satisfied by the Sobolev spacesW k,m (ω s) andW k,m(Ω), where {Ω s} is a sequence of perforated domains contained in a bounded regionΩ ⊂R n. Hence, the results obtained for abstract operators can be applied to the operators of the Dirichlet problem in the domainsΩ s.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. A. Marchenko and E. Ya. Khruslov,Boundary-Value Problems in the Domains with Fine-Grain Boundary [in Russian], Naukova Dumka, Kiev (1974).
E. Ya. Khruslov, “The first boundary-value problem in the domains with complex boundary for higher-order equations,”Mat. Sb.,103, No. 4, 614–629 (1977).
L. S. Pankratov,Asymptotic Behavior of Solutions to Variational Problems in the Domains with Complex Boundary, Preprint No. 11–87, Physical Engineering Institute of Low Temperatures, Ukrainian Academy of Sciences, Khar'kov (1987).
I. V. Skrypnik, “On the convergence of solutions to a nonlinear Dirichlet problem under refinement of the boundary,”Za. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov,115, 236–250 (1982).
I. V. Skrypnik, “A quasilinear Dirichlet problem for the domains with fine-grain boundary,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 2, 21–25 (1982).
I. V. Skrypnik, “Averaging of quasilinear elliptic problems in perforated domains,”Uspekhi Mat. Nauk,40, No. 4, 197–198 (1985).
I. V. Skrypnik, “Averaging of nonlinear Dirichlet problems in the domains with channels,”Dokl. Akad. Nauk SSSR,313, No. 5, 1049–1053 (1990).
I. V. Skrypnik and A. I. Prokopenko,Averaging of the Dirichlet Problem for Higher-Order Quasilinear Equations of Elliptic Type in the Domains with Channels [in Russian], Preprint No. 89.13, Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk (1989).
A. I. Prokopenko,The Dirichlet Problem for Higher-Order Nonlinear Equations in the Domains with Fine-Grain Boundary [in Russian], Dep. UkrNIINTI No. 599, Donetsk (1985).
V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, and Ha Tien Ngoan, “Averaging andG-convergence of differential operators,”Uspekhi Mat. Nauk,34, No. 5, 65–133 (1979).
J.-L. Lions,Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires [Russian translation], Mir, Moscow (1972).
A. A. Kovalevskii, “The G-convergence of abstract operators with different domains of definition,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 3, 20–23 (1989).
A. A. Kovalevskii,The G-Convergence and Averaging of Nonlinear Elliptic Operators with Different Domains of Definition [in Russian], Preprint No. 90.01, Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk (1990).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 948–962, July, 1993.
Rights and permissions
About this article
Cite this article
Kovalevskii, A.A. On theg-convergence of nonlinear elliptic operators related to the dirichlet problem in variable domains. Ukr Math J 45, 1049–1065 (1993). https://doi.org/10.1007/BF01057452
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01057452