Quasiconformal mappings and the rank of $\tfrac {dDf}{d|Df|}$ for $f\in BV(\mathbb{R}^n; \mathbb{R}^n)$

TL;DR

This paper introduces a relaxed distortion measure for BV functions and characterizes the full-rank condition of the derivative's Radon-Nikodym derivative in terms of this measure, advancing the understanding of quasiconformal mappings in BV spaces.

Contribution

It defines a relaxed distortion number for BV functions and establishes a precise equivalence between finite distortion and full rank of the derivative almost everywhere.

Findings

The relaxed distortion number $H_f^{ extrm{fine}}$ is finite almost everywhere if and only if the derivative has full rank.

The characterization links geometric distortion properties with the algebraic rank of the derivative.

Provides new insights into the structure of BV functions related to quasiconformal mappings.

Abstract

We define a relaxed version $H_f^{\textrm{fine}}$ of the distortion number $H_f$ that is used to define quasiconformal mappings. Then we show that for a BV function $f\in BV(\mathbb{R}^n;\mathbb{R}^n)$, for $|Df|$-a.e. $x\in\mathbb{R}^n$ it holds that $H_{f^*}^{\textrm{fine}}(x)<\infty$ if and only if $\tfrac{dDf}{d|Df|}(x)$ has full rank.