Ciarlet Ne\v{c}as condition in fractional Sobolev spaces

TL;DR

This paper investigates the Ciarlet-Nečas condition within fractional Sobolev spaces, demonstrating that a positive Jacobian and a change of variables formula imply almost-everywhere injectivity of deformations in fractional nonlinear elasticity.

Contribution

It extends the Ciarlet-Nečas condition to fractional Sobolev spaces and proves that this condition ensures almost-everywhere injectivity for deformations with positive Jacobian.

Findings

Change of variables formula in fractional Sobolev spaces.

Ciarlet-Nečas condition implies a.e. injectivity.

Positive Jacobian ensures deformation invertibility.

Abstract

Let $s\in(\frac{n}{n+1},1)$, $\Omega\subset\mathbb{R}^n$ be an open set and let $f\in W^{s,n/s}(\Omega,\mathbb{R}^n)$ be mapping with positive distributional Jacobian $\mathcal{J}_f>0$ which models some deformation in fractional Nonlinear Elasticity. We show change of variables formula in this class and as a consequence we show that the analogue of Ciarlet-Ne\v{c}as condition $\mathcal{J}_f(\Omega)=|f(\Omega)|$ implies that our mapping is one-to-one a.e.