Homogenization and integral representation of energy functionals in manifold valued Orlicz-Sobolev spaces

TL;DR

This paper extends integral representation results for homogenization and dimensional reduction to manifold-valued fields in Orlicz-Sobolev spaces, using $ ext{Gamma}$-convergence and growth conditions.

Contribution

It generalizes classical results to the Orlicz setting beyond Sobolev spaces, establishing $ ext{Gamma}$-convergence and tangential quasiconvexity under $ ext{Delta}_2$ and $ ext{Nabla}_2$ conditions.

Findings

Proves $ ext{Gamma}$-convergence in the Orlicz setting.

Establishes a cell formula for the density of the $ ext{Gamma}$-limit.

Shows the density is a tangential quasiconvex integrand.

Abstract

This paper aims to extend to Orlicz-Sobolev spaces some results of integral representation for the simultaneous homogenization and dimensional reduction of integral energies defined on fields taking values on a differentiable manifold. Since our functional framework goes beyond the classical Sobolev's spaces, we also prove, via $\Gamma$-convergence, a general integral representation results in the unconstrained Orlicz setting. Due to $\Delta_2$ and $\nabla_2$ conditions verified by the Young function $\Phi$ (which modulated the growth behaviour), we prove that the density of the $\Gamma$-limit is a tangential quasiconvex integrand represented by a cell formula.