A+ CATEGORY SCIENTIFIC UNIT

Weak limits of fractional Sobolev homeomorphisms are almost injective

Armin Schikorra, James M. Scott Studia Mathematica 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.

Authors

  • Armin SchikorraDepartment of Mathematics
    University of Pittsburgh
    Pittsburgh, PA 15260, USA
    e-mail
  • James M. ScottDepartment of Mathematics
    University of Pittsburgh
    Pittsburgh, PA 15260, USA
    e-mail

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