Invertibility of Sobolev mappings under minimal hypotheses
The paper deals with a version of the inverse function theorem for continuous weakly differentiable mappings \(f : \Omega \rightarrow {\mathbb R}^n\) with \(\Omega\) being a bounded domain in \({\mathbb R}^n\). The main result of the paper is the following Theorem. Suppose \(f\in W_{\text{loc}}^{1, n}(\Omega, {\mathbb R}^n\) is a nonconstant mapping such that the mean inner distortion \({\mathcal K}_{\Omega} [f] < \infty\). If there exists \(\delta > - 1\) such that \(D f\in {\mathcal M}(\delta)\) for almost every \(x\in \Omega\), then \(f\) is a local homeomorphism. The proof of this theorem is based on two results of independent interest. The first step consists in the proof that the mapping is discrete and open; that is, preimages of points are discrete sets, and images of open sets are open. Another ingredient of the proof is an estimate for the multiplicity \(N(y, f, A)\) \(:=\) \( \sharp \left(f^{-1}(y)\cap A\right)\) of a local homeomorphism \(f\) in terms of the integral of the inner distortion \({\mathcal K}_{I}(\cdot, f)\) in dimension \(n\geq 3\).
scientific article; zbMATH DE number 5690770
scientific article