Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

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 $p\le n-1$ 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.