Values of finite distortion: Reshetnyak's theorem and the Lusin (N) -property

TL;DR

This paper extends Reshetnyak's theorem to mappings with finite distortion values, establishing discreteness and topological properties of preimages under certain integrability conditions.

Contribution

It introduces a unified framework for finite distortion and quasiregular values, proving a single-value analogue of Reshetnyak's theorem with new integrability conditions.

Findings

Preimages of finite distortion values are discrete under specified conditions.

Mappings with certain distortion inequalities preserve measure-zero sets.

The local topological index is positive at preimage points.

Abstract

Let $\Omega \subset \mathbb{R}^n$ be a domain and $f \in W^{1,n}_{\text{loc}} (\Omega,\mathbb{R}^n) $. We say that $f$ has a value of finite distortion at $y_0 \in \mathbb{R}^n$ if there exist measurable functions $K \colon \Omega \to [0,\infty) $ and $\Sigma \in L^1_{\text{loc}} (\Omega)$ such that \[ \lvert Df(x)\rvert^n \le K(x) \det Df (x) + \Sigma(x) \lvert f(x)-y_0 \rvert^n \quad \text{for a.e. } x \in \Omega. \] This notion unifies the classical theory of mappings of finite distortion with the recently introduced theory of quasiregular values. We establish a single-value analogue of Reshetnyak's theorem in this setting. Specifically, if $f$ is nonconstant and has a value of finite distortion at $y_0$, with $K \in L^p_{\text{loc}}(\Omega) $, $\Sigma/K \in L^q_{\text{loc}}(\Omega)$, $p>n-1$, and $p^{-1}+q^{-1}<1$, then the preimage $f^{-1}\{y_0\}$ is discrete, the local topological index is positive at every point of $f^{-1}\{y_0\}$, and every neighborhood of a point in $f^{-1}\{y_0\}$ is mapped onto a neighborhood of $y_0$. We also prove that mappings satisfying a more general distortion inequality with defect preserve sets of Lebesgue measure zero.