The $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions

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.
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