Weak Limits of Fractional Sobolev Homeomorphisms are Almost Injective: A Note

Let $Omega subset mathbb{R}^n$ be an open set and $f_k in W^{s,p}(Omega;mathbb{R}^n)$ be a sequence of homeomorphisms weakly converging to $f in W^{s,p}(Omega;mathbb{R}^n)$. It is known that if $s=1$ and $p > n-1$ then $f$ is injective almost everywhere in the domain and the target. In this note we extend such results to the case $sin(0,1)$ and $sp > n-1$. This in particular applies to $C^s$-Holder maps.