Overview
This research investigates the notion of maps exhibiting a value of finite distortion within a domain $\Omega \subset \mathbb{R}^n$. Specifically, for a function $f \in W^{1,n}_{\text{loc}} (\Omega,\mathbb{R}^n)$, it defines the presence of a finite distortion value at $y_0 \in \mathbb{R}^n$ if measurable functions $K \colon \Omega \to [0,\infty)$ and $\Sigma \in L^1_{\text{loc}} (\Omega)$ exist, satisfying the inequality: $ \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 definition aims to unify the established theory of finite distortion mappings with the more recently introduced theory of quasiregular values.
Research Context
The study operates at the intersection of classical mathematical analysis concerning mappings of finite distortion and newer theoretical developments regarding quasiregular values. These two frameworks are distinct, and the proposed concept of a value of finite distortion serves as a unifying element. The research builds upon foundational theorems in geometric function theory, specifically Reshetnyak's theorem and the Liouville theorem, extending their applicability to this unified context.
Approach
The research establishes theoretical results based on the defined distortion inequality. The methodology involves demonstrating single-value analogues of Reshetnyak's theorem and the Liouville theorem. This is achieved under what are described as 'sharp integrability assumptions' on the functions $K$ and $\Sigma$. Additionally, the study investigates the Lusin (N)-property, aiming to prove that mappings which satisfy a more generalized distortion inequality, referred to as an inequality 'with defect,' consistently preserve sets of Lebesgue measure zero.
Findings
- The concept of a value of finite distortion at $y_0 \in \mathbb{R}^n$ unites the classical theory of mappings of finite distortion and the theory of quasiregular values.
- Single-value analogues of Reshetnyak's theorem were established.
- Single-value analogues of the Liouville theorem were established.
- These single-value analogues are valid under sharp integrability assumptions on the functions $K$ and $\Sigma$.
- Mappings satisfying a more general distortion inequality with defect preserve sets of Lebesgue measure zero.