# Homogenization of nonlinear elliptic systems in nonreflexive Musielak-Orlicz spaces

## TL;DR

This paper investigates the homogenization of nonlinear elliptic systems within general Musielak-Orlicz spaces, extending results beyond classical growth conditions and accommodating spatially dependent anisotropic N-functions.

## Contribution

It develops a homogenization framework for elliptic systems in nonreflexive Musielak-Orlicz spaces without requiring $ riangle_2$ and $
abla_2$ conditions, broadening applicability.

## Key findings

- Homogenization results hold without $ riangle_2$ and $
abla_2$ conditions.
- The approach handles spatially dependent anisotropic N-functions.
- The work extends classical homogenization theory to more general function spaces.

## Abstract

We study the homogenization process for families of strongly nonlinear elliptic systems with the homogeneous Dirichlet boundary conditions. The growth and the coercivity of the elliptic operator is assumed to be indicated by a general inhomogeneous anisotropic $N-$function, which may be possibly also dependent on the spatial variable, i.e., the homogenization process will change the characteristic function spaces at each step. Such a problem is well known and there exists many positive results for the function satisfying $\Delta_2$ and $\nabla_2$ conditions an being in addition H\"{o}lder continuous with respect to the spatial variable. We shall show that cases these conditions can be neglected and will deal with a rather general problem in general function space setting.