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. Under sharp integrability assumptions on $K$ and $\Sigma$, we establish single-value analogues of Reshetnyak's theorem and the Liouville theorem. We also prove that mappings satisfying a more general distortion inequality with defect preserve sets of Lebesgue measure zero.
From: Ilmari Kangasniemi [view email]
[v1]
Tue, 3 Feb 2026 08:38:54 UTC (25 KB)
[v2]
Fri, 29 May 2026 12:04:32 UTC (32 KB)
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