Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

TL;DR

This paper establishes the $ ext{Gamma}$-limit of a family of integral functionals with nonstandard growth in an Orlicz Sobolev setting, considering multiscale homogenization for second-order derivatives.

Contribution

It provides a novel multiscale homogenization result for integral convex functionals with nonstandard growth in an Orlicz Sobolev space, including characterization of second-order derivative limits.

Findings

$ ext{Gamma}$-limit derived for multiscale functionals

Characterization of reiterated two-scale limits of second derivatives

Results applicable to convex integrands with nonstandard growth

Abstract

The $\Gamma $-limit of a family of functionals $u\mapsto \int_{\Omega }f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx$ 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.