Geodesic
$G$-compactness of sequences of non-linear operators of Dirichlet problems with a~variable domain of definition
Izvestiya. Mathematics , Tome 60 (1996) no. 1, pp. 137-168.

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For a sequence of operators $A_s\colon\overset{\circ}{W}{}^{1,m}(\Omega_s)\to\bigl(\overset{\circ}{W}{}^{1,m}(\Omega_s)\bigr)^*$ in divergence form we prove a theorem concerning the choice of a subsequence that $G$-converges to the operator $\widehat A\colon\overset{\circ}{W}{}^{1,m}(\Omega)\to\bigl(\overset{\circ}{W}{}^{1,m}(\Omega)\bigr)^*$ 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. 
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A. A. Kovalevsky. $G$-compactness of sequences of non-linear operators of Dirichlet problems with a~variable domain of definition. Izvestiya. Mathematics , Tome 60 (1996) no. 1, pp. 137-168. http://geodesic.mathdoc.fr/item/IM2_1996_60_1_a5/

[1] Kovalevsky A. A., “On $G$-convergence of operators of Dirichlet problem with varying domain”, Dokl. Akad. Nauk Ukr., 1993, no. 5, 13–17 | MR

[2] Kovalevskii A. A., “O $G$-skhodimosti nelineinykh ellipticheskikh operatorov, svyazannykh s zadachei Dirikhle v peremennykh oblastyakh”, Ukr. matem. zhurn., 45:7 (1993), 948–962 | MR | Zbl

[3] Khruslov E. Ya., “Pervaya kraevaya zadacha v oblastyakh so slozhnoi granitsei dlya uravnenii vysshikh poryadkov”, Matem. sb., 103:4 (1977), 614–629 | MR | Zbl

[4] Skrypnik I. V., “Kvazilineinaya zadacha Dirikhle dlya oblastei s melkozernistoi granitsei”, DAN USSR. Ser. A, 1982, no. 2, 21–25 | MR | Zbl

[5] Skrypnik I. V., Metody issledovaniya nelineinykh ellipticheskikh granichnykh zadach, Nauka, M., 1990 | MR

[6] Pankratov L. S., Ob asimptoticheskom povedenii reshenii variatsionnykh zadach v oblastyakh so slozhnoi granitsei, Preprint No 242 11-87, FTINT AN USSR, Kharkov, 1987

[7] Dal Maso G., Garroni A., A new approach to the study of limits of Dirichlet problems in perforated domains, Preprint No 242 67/M, SISSA, Trieste, 1993 | Zbl

[8] Dal Maso G., Murat F., Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Preprint, SISSA, Trieste, 1993 | Zbl

[9] Zhikov V. V., “O perekhode k predelu v nelineinykh variatsionnykh zadachakh”, Matem. sb., 183:8 (1992), 47–84 | MR

[10] Kovalevsky A. A., On $G$-compactness of a sequence of nonlinear operators of Dirichlet problems in variable domains, Preprint No 242 93.09, IAMM, Donetsk, 1993 | MR

[11] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[12] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, 4-e izd., pererab., Nauka, M., 1976 | MR