Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

Abstract

The $\Gamma$-limit of a family of functionals $u\mapsto {\int }_{\Omega }f\left(\frac{x}{\epsilon },\frac{x}{{\epsilon }^2},D^{s}u\right)dx$ is obtained for $s=1,2$ and when the integrand $f=f\left(y,z,v\right)$ is a continous function, periodic in $y$ and $z$ and convex with respect to $v$ with nonstandard growth. The reiterated two-scale limits of second order derivative are characterized in this setting.