Bilipschitz mappings with derivatives of bounded variation
Let \(\Omega\subset \mathbb{R}^n\) be an open set. A function \(h\in L^1(\Omega)\) is of bounded variation, \(h\in BV(\Omega)\), if the distributional partial derivatives of \(h\) are measures with finite total variation in \(\Omega\). \(BV(\Omega, \mathbb{R}^m)\) is defined as the class of all \(f\in L^1(\Omega,\mathbb{R}^m)\) the coordinate functions of which belong to \(BV(\Omega)\). \(BV_{\text{loc}}(\Omega,\mathbb{R}^m)\) is the class of all \(f:\Omega\to \mathbb{R}^m\) for which \(f\in BV(\Omega',\mathbb{R}^m)\) for each open \(\Omega'\subset \Omega\). Analogously, the Sobolev space \(W_{\text{loc}}^{1,p}(\Omega,\mathbb{R}^m)\) is defined. In the paper the following theorems are proved. Theorem 1. Let \(\Omega,\Omega'\subset \mathbb{R}^n\) be open and suppose that \(f:\Omega\to \Omega\) is a bilipschitz mapping such that \(Df\in BV_{\text{loc}}(\Omega, \mathbb{R}^{n^2})\). Then \(Df^{-1}\in BV_{\text{loc}}(\Omega',\mathbb{R}^{n^2})\). Theorem 2. Let \(\Omega,\Omega'\subset \mathbb{R}^n\) be open, \(p\geq 1\) and suppose that \(f:\Omega\to \Omega'\) is a bilipschitz mapping such that \(Df\in W_{\text{loc}}^{1,p}(\Omega, \mathbb{R}^{n^2})\). Then \(Df^{-1}\in W_{\text{loc}}^{1,p}(\Omega',\mathbb{R}^{n^2})\). An example is given which shows that Theorem 1 is not valid in dimension \(n\geq 4\) without the assumption that \(f\) is Lipschitz.
scientific article; zbMATH DE number 5302054