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

Abstract

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 $s\in (0,1)$ and $sp>n-1$. This in particular applies to $C^s$-Hölder maps.