Bilipschitz mappings with derivatives of bounded variation

Abstract

Let $\Omega\subset\mathbb{R}^n$ be open and suppose that $f\colon \Omega\to\mathbb{R}^n$ is a bilipschitz mapping such that $Df\in BV_{\operatorname{loc}}(\Omega,\mathbb{R}^{n^2})$. We show that under these assumptions the inverse satisfies $Df^{-1}\in BV_{\operatorname{loc}}(f(\Omega),\mathbb{R}^{n^2})$.