Geodesic
Homogenization of variational inequalities and equations
Sbornik. Mathematics, Tome 199 (2008) no. 1, pp. 67-98 Cet article a éte moissonné depuis la source Math-Net.Ru

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Results on the convergence of sequences of solutions of non-linear equations and variational inequalities for obstacle problems are proved. The variational inequalities and equations are defined by a non-linear, pseudomonotone operator of the second order with periodic, rapidly oscillating coefficients and by sequences of functions characterizing the obstacles and the boundary conditions. Two-scale and macroscale (homogenized) limiting problems for such variational inequalities and equations are obtained. Results on the relationship between solutions of these limiting problems are established and sufficient conditions for the uniqueness of solutions are presented. Bibliography: 25 titles. 
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G. V. Sandrakov. Homogenization of variational inequalities and equations. Sbornik. Mathematics, Tome 199 (2008) no. 1, pp. 67-98. http://geodesic.mathdoc.fr/item/SM_2008_199_1_a3/

[1] G. V. Sandrakov, “Homogenization of variational inequalities for problems with a regular obstacle”, Russian Acad. Sci. Dokl. Math., 70:1 (2004), 539–542 | MR

[2] G. V. Sandrakov, “Homogenization of variational inequalities for obstacle problems”, Sb. Math., 196:4 (2005), 541–560 | DOI | MR | Zbl

[3] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, de Gruyter, Paris, 1969 | MR | MR | Zbl | Zbl

[4] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, Stud. Math. Appl., 5, North-Holland, Amsterdam–New York–Oxford, 1978 | MR | Zbl

[5] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Pure Appl. Math., 88, Academic Press, New York–London, 1980 | MR | MR | Zbl | Zbl

[6] A. Mokrane, F. Murat, “The Lewy–Stampacchia inequality for bilateral problems”, Ricerche Mat., 53:1 (2004), 139–182 | MR | Zbl

[7] G. Allaire, “Homogenization and two-scale convergence”, SIAM J. Math. Anal., 23:6 (1992), 1482–1518 | DOI | MR | Zbl

[8] I. Ekeland, R. Temam, Convex analysis and variational problems, Stud. Math. Appl., 1, North-Holland, Amsterdam–Oxford; Elsevier, New York, 1976 | MR | MR | Zbl | Zbl

[9] M. Valadier, “Two-scale convergence and Young measures”, Prepublication du Laboratoire d'analyse convexe, Universite Montpellier II, Montpellier, 1995

[10] A. A. Amosov, “Weak convergence for a class of rapidly oscillating functions”, Math. Notes, 62:1 (1997), 122–126 | DOI | MR | Zbl

[11] S. Fučik, A. Kufner, Nonlinear differential equations, Stud. Appl. Mech., 2, Elsevier, Amsterdam–New York, 1980 | MR | MR | Zbl | Zbl

[12] G. Nguetseng, “A general convergence result for a functional related to the theory of homogenization”, SIAM J. Math. Anal., 20:3 (1989), 608–623 | DOI | MR | Zbl

[13] N. Fusco, G. Moscariello, “On the homogenization of quasilinear divergence structure operators”, Ann. Mat. Pura Appl. (4), 146:1 (1986), 1–13 | DOI | MR | Zbl

[14] N. Bakhvalov, G. Panasenko, Homogenisation: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials, Math. Appl. (Soviet Ser.), 36, Kluwer Acad. Publ., Dordrecht, 1989 | MR | MR | Zbl | Zbl

[15] P. Chiadò Piat, A. Defranceschi, “Homogenization of monotone operators”, Nonlinear Anal., 14:9 (1990), 717–732 | DOI | MR | Zbl

[16] G. V. Sandrakov, “Homogenization of nonlinear equations and variational inequalities with obstacles”, Russian Acad. Sci. Dokl. Math., 73:2 (2006), 178–181 | MR | MR

[17] N. Dunford, J. T. Schwartz, Linear operators. I. General theory., Pure Appl. Math., 7, Intersci. Publ., New York–London, 1958 | MR | MR | Zbl

[18] H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Academie-Verlag, Berlin, 1974 | MR | MR | Zbl

[19] V. V. Zhikov, “On two-scale convergence”, J. Math. Sci. (N. Y.), 120:3 (2004), 1328–1352 | DOI | MR | Zbl

[20] D. Lukkassen, P. Wall, “Two-scale convergence with respect to measures and homogenization of monotone operators”, J. Funct. Spaces Appl., 3:2 (2005), 125–161 | MR | Zbl

[21] V. V. Zhikov, M. E. Rychago, S. B. Shul'ga, “Homogenization of monotone operators by the method of two-scale convergence”, J. Math. Sci. (N. Y.), 127:5 (2005), 2159–2173 | DOI | MR | Zbl

[22] J. Casado-Díaz, I. Gayte, “The two-scale convergence method applied to generalized Besicovitch spaces”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458:2028 (2002), 2925–2946 | DOI | MR | Zbl

[23] V. Chiadò Piat, F. Serra Cassano, “Some remarks about the density of smooth functions in weighted Sobolev spaces”, J. Convex Anal., 1:2 (1994), 135–142 | MR | Zbl

[24] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and quasilinear elliptic equations, Math. Sci. Engrg., 46, Academic Press, New York–London, 1968 | MR | MR | Zbl | Zbl

[25] R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart Winston, New York–Toronto–London, 1965 | MR | Zbl | Zbl