On integral representation of $\Gamma$-limit functionals

We consider the $\Gamma$-convergence of a sequence of integral functionals $F_n(u)$, defined on the functions $u$ from the Sobolev space $W^{1,\alpha}(\Omega)$ ($\alpha>1$), $\Omega$ is a bounded Lipschitz domain, where the integrand $f_n(x,u,\nabla u)$ depends on a function $u$ and its gradient $\nabla u$. As functions of $\xi$, the integrands $f_n(x,s,\xi)$ are convex and satisfy a two-sided power estimate on the coercivity and growth with different exponents $\alpha<\beta$. Besides, the integrands $f_n(x,s,\xi)$ are equi-continuous over $s$ in some sense with respect to $n$. We prove that for the functions from $L^\infty(\Omega)\cap W^{1,\beta}(\Omega)$ the $\Gamma$-limit functional coincides with an integral functional $F(u)$ for which the integrand $f(x,s,\xi)$ is of the same class as $f_n(x,s,\xi)$.