Ciarlet Nečas 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-Nečas condition ${\mathcal{J}}_f(\Omega )=|f(\Omega )|$ implies that our mapping is one-to-one a.e.