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

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a domain and $f\in W_{\text{loc}}^{1,n}(\Omega ,{\mathbb{R}}^n)$. We say that $f$ has a value of finite distortion at $y_0\in {\mathbb{R}}^n$ if there exist measurable functions $K:\Omega \to [0,\infty )$ and $\Sigma \in L_{\text{loc}}^1(\Omega )$ 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. 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.

34 pages; the updated version adds the Liouville theorem of values of finite distortion, along with other significant refinements and improvements