Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

Abstract

The Γ-limit of a family of functionals u → Ω f x ε , x ε 2 , D s u dx is obtained for s = 1, 2 and when the integrand f = f (y, z, v) 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 derivatives are characterized in this setting.

Some convergence results on the periodic Unfolding Operator in Orlicz setting

“…Homogenization of periodic structures via two-scale convergence in Orlicz setting was introduced in [13] and later expanded in [14,15,16] in order to deal with convex integral functionals, multiscale problems, and differential operators.…”

“…To this end, in subsection 1.2 we deal with preliminaries on Orlicz spaces, while the remaining of the paper is devoted to define the unfolding method in this framework and prove some convergence results, establishing parallels with the strong and weak convergence in L B (Ω × Y ) and L B (Ω ) between unfolded sequences and their original counterparts, thus extending to the Orlicz setting the result proved by [5,6,7,8], leaving the parallel with the notion of two scale convergence in the standard and Orlicz setting (cf. [21,2,18,22,13]), as well as the applications, the extension to more scales and to higher order derivatives as in [3,12,19,14,15,16] for a forthcoming paper.…”