Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

TL;DR

This paper proves that weak limits of Sobolev homeomorphisms are almost everywhere injective when the integrability exponent exceeds n-1, and constructs counterexamples for lower exponents.

Contribution

It establishes almost everywhere injectivity of weak limits of Sobolev homeomorphisms for p > n-1 and provides explicit counterexamples for p ≤ n-1.

Findings

Weak limits are injective a.e. for p > n-1.

Counterexamples show non-injectivity for p ≤ n-1.

Preimages can be continua with positive measure.

Abstract

Let $\Omega\subset\mathbb{R}^n$ be an open set and let $f\in W^{1,p}(\Omega,\mathbb{R}^n)$ be a weak (sequential) limit of Sobolev homeomorphisms. Then $f$ is injective almost everywhere for $p>n-1$ both in the image and in the domain. For $p\leq n-1$ we construct a strong limit of homeomorphisms such that the preimage of a point is a continuum for every point in a set of positive measure in the image and a topological image of a point is a continuum for every point in a set of positive measure in the domain.