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Injectivity almost everywhere for weak limits of Sobolev homeomorphisms. (English) Zbl 1446.30040

Summary: 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 the topological image of a point is a continuum for every point in a set of positive measure in the domain.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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