  This article is devoted to obtain the $\Gamma$-limit, as $\epsilon$ tends to
zero, of the family of functionals
$$F_{\epsilon}(u)=\int_{\Omega}f\Bigl(x,\frac{x}{\epsilon},...,
\frac{x}{\epsilon^n},\nabla u(x)\Bigr)dx$$, where $f=f(x,y^1,...,y^n,z)$ is
periodic in $y^1,...,y^n$, convex in $z$ and satisfies a very weak regularity
assumption with respect to $x,y^1,...,y^n$. We approach the problem using the
multiscale Young measures.
