Accepted Paper

Inserted: 3 jan 2019
Last Updated: 26 sep 2023

Journal: Nonlinear Analysis
Volume: 194
Number: May 2020
Year: 2020
Doi: https://doi.org/10.1016/j.na.2019.04.012

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.

Keywords: Topological degree, Orlicz , Weak monotonicity