Homogenization and 3D-2D dimension reduction of a functional on manifold valued Sobolev spaces

TL;DR

This paper investigates the combined process of homogenization and dimensional reduction for functionals on manifold-valued Sobolev spaces, establishing a cell formula for the limit's density in a superlinear growth setting.

Contribution

It introduces a novel approach to analyze the $ ext{Gamma}$-limit of integral functionals on manifold-valued Sobolev spaces, providing a cell formula for the density.

Findings

The $ ext{Gamma}$-limit density is tangential quasiconvex.

A cell formula characterizes the limit integrand.

Results apply to superlinear growth regimes.

Abstract

We study simultaneous homogenization and dimensional reduction of integral functionals for maps in manifold-valued Sobolev spaces. Due to the superlinear growth regime, we prove that the density of the $\Gamma$-limit is a tangential quasiconvex integrand represented by a cell formula.