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

TL;DR

This paper extends known results about the almost injectivity of weak limits of Sobolev homeomorphisms from the classical case to fractional Sobolev spaces, showing that under certain conditions, the limit map retains almost injectivity.

Contribution

It generalizes the almost injectivity property of Sobolev homeomorphisms to fractional Sobolev spaces for s in (0,1) with sp > n-1, broadening the applicability to Hölder maps.

Findings

Weak limits of fractional Sobolev homeomorphisms are almost injective under specified conditions.

The result applies to $C^s$-Hölder maps, extending classical injectivity results.

The paper establishes conditions for the preservation of injectivity properties in fractional Sobolev spaces.

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 $C^s$-H\"older maps.