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arXiv · 1912.05413

Injectivity almost everywhere for weak limits of Sobolev homeomorphisms

Ondřej Bouchala·Stanislav Hencl·Anastasia Molchanova
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Abstract

Let $Ω\subset\mathbb{R}^n$ be an open set and let $f\in W^{1,p}(Ω,\mathbb{R}^n)$ 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.

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Ondřej Bouchala, Stanislav Hencl, Anastasia Molchanova. 2019-12-11. Injectivity almost everywhere for weak limits of Sobolev homeomorphisms. https://arxiv.org/abs/1912.05413

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