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

中文翻译：

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索博列夫同胚弱极限的可射性几乎无处不在

让 $\Omega\subset\mathbb{R}^n$ 是一个开集，让 $f\in W^{1,p}(\Omega,\mathbb{R}^n)$ 是一个弱（序列）索博列夫同胚的极限。然后 $f$ 在图像和域中几乎处处都是单射的，因为 $p>n-1$ 。对于 $p\leq n-1$ 我们构造了同胚的强极限，使得点的原像是图像中一组正测度中每个点的连续统，而一个点的拓扑图像是每个点的连续统指向域中的一组正度量。