Weak limits of fractional Sobolev homeomorphisms are almost injective
Volume 269 / 2023
Studia Mathematica 269 (2023), 241-260
MSC: Primary 46E35; Secondary 47H11.
DOI: 10.4064/sm201218-20-9
Published online: 17 November 2022
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 $p \gt n-1$ 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 $sp \gt n-1$. This in particular applies to $C^s$-Hölder maps.
