# Orlicz-Sobolev nematic elastomers

## TL;DR

This paper extends existence theorems for nematic elastomers and magnetoelasticity models to Orlicz spaces, allowing for broader energy density classes while maintaining key regularity and invertibility properties of deformation maps.

## Contribution

It generalizes previous models by incorporating Orlicz-Sobolev spaces, enabling analysis of more complex energy densities with minimal regularity assumptions.

## Key findings

- Established compactness and lower semicontinuity in Orlicz spaces.
- Proved regularity and invertibility properties of deformation maps in the Orlicz setting.
- Extended existence theorems to a larger class of energy densities.

## Abstract

We extend the existence theorems in [Barchiesi, Henao \& Mora-Corral; ARMA 224], for models of nematic elastomers and magnetoelasticity, to a larger class in the scale of Orlicz spaces. These models consider both an elastic term where a polyconvex energy density is composed with an unknown state variable defined in the deformed configuration, and a functional corresponding to the nematic energy (or the exchange and magnetostatic energies in magnetoelasticity) where the energy density is integrated over the deformed configuration. In order to obtain the desired compactness and lower semicontinuity, we show that the regularity requirement that maps create no new surface can still be imposed when the gradients are in an Orlicz class with an integrability just above the space dimension minus one. We prove that the fine properties of orientation-preserving maps satisfying that regularity requirement (namely, being weakly 1-pseudomonotone, $\mathcal H^1$-continuous, a.e.\ differentiable, and a.e.\ locally invertible) are still valid in the Orlicz-Sobolev setting.