Lipschitz maps and nets in Euclidean space. (English) Zbl 0941.37030

This paper deals with the following three questions:
1. Is there a bi-Lipschitz homeomorphism \(\varphi:\mathbb{R}^n\to\mathbb{R}^n\) such that the Jacobian determinant \(\text{det }D\varphi=f\), where \(f\) is a given real-valued function \(f\in L^\infty(\mathbb{R}^n)\) with \(\inf f(x)>0\)?
2. Is there a Lipschitz or quasiconformal vector field with \(\text{div }v=f\), where \(f\) is a given \(f\in L^\infty(\mathbb{R}^n)\)?
3. Given a separated net \(Y\subset \mathbb{R}^n\), is there a bi-Lipschitz map \(\varphi:Y\to\mathbb{Z}^n\)?
Note that in the case of \(n=1\) all three questions have a positive answer. The author shows that for \(n>1\) the answer to all three questions is negative. He also proves that all three questions have positive solutions if the Lipschitz condition is replaced by a Hölder condition.


37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37F05 Dynamical systems involving relations and correspondences in one complex variable
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