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

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.