Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

Abstract

Let Ω ⊂ R n be an open set and let f ∈ W 1,p (Ω, 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 ≤ 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.

Global invertibility for orientation-preserving Sobolev maps via invertibility on or near the boundary

“…By contrast, this does not work for AI loc (Ω) as defined here. However, approximate invertibility defined with respect to weak convergence in the Sobolev space is a still meaningful concept [6].…”