Geodesic
The $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions
Sbornik. Mathematics, Tome 205 (2014) no. 4, pp. 488-521 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the $\Gamma$-convergence as $\varepsilon\to 0$ of a family of integral functionals with integrand $f_\varepsilon(x,u,\nabla u)$, where the integrand oscillates with respect to the space variable $x$. The integrands satisfy a two-sided power estimate on the coercivity and growth with different exponents. As a consequence, at least two different variational Dirichlet problems can be connected with the same functional. This phenomenon is called Lavrent'ev's effect. We introduce two versions of $\Gamma$-convergence corresponding to variational problems of the first and second kind. We find the $\Gamma$-limit for the aforementioned family of functionals for problems of both kinds; these may be different. We prove that the $\Gamma$-convergence of functionals goes along with the convergence of the energies and minimizers of the variational problems. Bibliography: 23 titles.
Keywords: homogenization, $\Gamma$-realizing sequence, upper and lower regularization.
Mots-clés : $\Gamma$-convergence, Lavrent'ev's effect 
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V. V. Zhikov; S. E. Pastukhova. The $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions. Sbornik. Mathematics, Tome 205 (2014) no. 4, pp. 488-521. http://geodesic.mathdoc.fr/item/SM_2014_205_4_a2/

[1] E. De Giorgi, T. Franzoni, “Su un tipo di convergenza variazionale”, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58:6 (1975), 842–850 | MR | Zbl

[2] E. De Giorgi, G. Letta, “Une notion générale de convergence faible des fonctions croissantes d'ensemble”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4:1 (1977), 61–99 | MR | Zbl

[3] E. De Giorgi, “Sulla convergenza di alcune successioni di integrali del tipo dell'area”, Rend. Mat. (6), 8 (1975), 277–294 | MR | Zbl

[4] L. Carbone, C. Sbordone, “Some properties of $\Gamma$-limits of integral functionals”, Ann. Mat. Pura Appl. (4), 122:1 (1979), 1–60 | DOI | MR | Zbl

[5] V. V. Zhikov, “Questions of convergence, duality, and averaging for functionals of the calculus of variations”, Math. USSR-Izv., 23:2 (1984), 243–276 | DOI | MR | Zbl

[6] V. V. Zhikov, “On passage to the limit in nonlinear variational problems”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 427–459 | DOI | MR | Zbl

[7] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Usrednenie differentsialnykh operatorov, Nauka, M., 1993, 464 pp. | MR | Zbl

[8] V. V. Jikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994, xii+570 pp. | DOI | MR | Zbl

[9] G. Dal Maso, An introduction to $\Gamma$-convergence, Progr. Nonlinear Differential Equations Appl., 8, Birkhäuser Boston, Inc., Boston, MA, 1993, xiv+340 pp. | DOI | MR | Zbl

[10] A. Braides, A. Defranceschi, Homogenization of multiple integrals, Oxford Lecture Ser. Math. Appl., 12, The Clarendon Press, Oxford Univ. Press, New York, 1998, xiv+298 pp. | MR | Zbl

[11] A. Braides, $\Gamma$-convergence for beginners, Oxford Lecture Ser. Math. Appl., 22, Oxford Univ. Press, Oxford, 2002, xii+218 pp. | MR | Zbl

[12] K. Kuratowski, Topology, v. I, new edition, revised and augmented, Academic Press, New York–London; Państwowe Wydawnictwo Naukowe, 1966, xx+560 pp. | MR | MR | Zbl | Zbl

[13] V. V. Zhikov, “The Lavrent'ev effect and averaging of nonlinear variational problems”, Differential Equations, 27:1 (1991), 32–39 | MR | Zbl

[14] V. V. Zhikov, “On variational problems and nonlinear elliptic equations with nonstandard growth conditions”, J. Math. Sci. (N. Y.), 173:5 (2011), 463–570 | DOI | MR | Zbl

[15] E. Weinan, “A class of homogenization problems in the calculus of variations”, Comm. Pure Appl. Math., 44:7 (1991), 733–759 | DOI | MR | Zbl

[16] V. V. Zhikov, S. E. Pastukhova, “$\Gamma$-convergence of integrands with nonstandard coercivity and growth conditions”, J. Math. Sci. (N. Y.), 196:4 (2014), 535–566 | DOI

[17] V. V. Zhikov, S. E. Pastukhova, “On the compensated compactness principle”, Dokl. Math., 82:1 (2010), 590–595 | DOI | MR | Zbl

[18] I. Ekeland, R. Temam, Convex analysis and variational problems, Stud. Math. Appl., 1, North-Holland Publishing Co., Amsterdam–Oxford; American Elsevier Publishing Co., Inc., New York, 1976, ix+402 pp. | MR | MR | Zbl | Zbl

[19] V. V. Jikov, “On Lavrentiev's phenomenon”, Russian J. Math. Phys., 3:2 (1995), 249–269 | MR | Zbl

[20] M. Giaquinta, E. Giusti, “On the regularity of the minima of variational integrals”, Acta Math., 148:1 (1982), 31–46 | DOI | MR | Zbl

[21] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. of Math. Stud., 105, Princeton Univ. Press, Princeton, NJ, 1983, vii+296 pp. | MR | Zbl

[22] A. D. Ioffe, “On lower semicontinuity of integral fuctionals. I”, SIAM J. Control Optim., 15:4 (1977), 521–538 | DOI | MR | Zbl

[23] M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, Ergeb. Math. Grenzgeb. (3), 34, 4th ed., Springer-Verlag, Berlin, 2008, xx+302 pp. | DOI | MR | Zbl