Values of finite distortion: Reshetnyak's theorem and the Lusin (N) -property
Ilmari Kangasniemi, Jani Onninen, Yizhe Zhu
TL;DR
The paper develops a single-value Reshetnyak-type theorem for mappings $f\in W^{1,n}_{\text{loc}}(\Omega,\mathbb{R}^n)$ that have a value of finite distortion at $y_0$ with data $(K,\Sigma)$, under the integrability condition $K\in L^p_{\text{loc}}(\Omega)$ and $\Sigma/K\in L^q_{\text{loc}}(\Omega)$ with $p>n-1$ and $p^{-1}+q^{-1}<1$. It introduces and exploits the spherical logarithm, weighted Jacobian analysis, and a degree-theoretic framework to show that either $f\equiv y_0$ or $f^{-1}\{y_0\}$ is discrete with positive local index, while every neighborhood of a preimage maps onto a neighborhood of $y_0$. A key contribution is the Lusin (N) property for solutions of $(K,\Sigma)$-distortion and value-of-finite-distortion maps in this regime, derived via refined oscillation and capacity estimates and almost weak monotonicity. The results provide a robust higher-dimensional analogue to Reshetnyak’s theorem for a broad class of nonlinear distortions and clarify the necessity of the integrability regime, with counterexamples in the planar and borderline settings. Overall, the work advances the understanding of open- and discrete-mapping behavior for non-uniform distortion and deepens the single-value quasiregular/finite-distortion theory with a concrete Lusin (N) result.
Abstract
Let $Ω\subset \mathbb{R}^n$ be a domain and $f \in W^{1,n}_{\text{loc}} (Ω,\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 Ω\to [0,\infty) $ and $Σ\in L^1_{\text{loc}} (Ω)$ such that \[ \lvert Df(x)\rvert^n \le K(x) \det Df (x) + Σ(x) \lvert f(x)-y_0 \rvert^n \quad \text{for a.e. } x \in Ω. \] 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}}(Ω) $, $Σ/K \in L^q_{\text{loc}}(Ω)$, $p>n-1$, 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.