Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

Abstract

Let Ω⊂Rn be an open set and let f∈W1,p(Ω,Rn) 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≤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 the topological image of a point is a continuum for every point in a set of positive measure in the domain.