Geodesic
The $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions
Sbornik. Mathematics, Tome 205 (2014) no. 4, pp. 488-521

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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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     author = {V. V. Zhikov and S. E. Pastukhova},
     title = {The $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions},
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     publisher = {mathdoc},
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     number = {4},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2014_205_4_a2/}
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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/