On the heterogeneous distortion inequality

TL;DR

This paper investigates Sobolev mappings satisfying a heterogeneous distortion inequality, establishing Liouville-type theorems and sharp H"older continuity estimates under certain integrability conditions on the distortion function, extending quasiregular mapping theory.

Contribution

It extends the theory of quasiregular mappings to more general solutions satisfying a heterogeneous distortion inequality, providing new regularity results and answering a longstanding open question.

Findings

Liouville-type theorem for solutions with heterogeneous distortion

Sharp H"older continuity estimates under integrability conditions

Extension of quasiregular mapping theory to broader class of solutions

Abstract

We study Sobolev mappings $f \in W_{\mathrm{loc}}^{1,n} (\mathbb{R}^n, \mathbb{R}^n)$, $n \ge 2$, that satisfy the heterogeneous distortion inequality \[\left|Df(x)\right|^n \leq K J_f(x) + \sigma^n(x) \left|f(x)\right|^n\] for almost every $x \in \mathbb{R}^n$. Here $K \in [1, \infty)$ is a constant and $\sigma \geq 0$ is a function in $L^n_{\mathrm{loc}}(\mathbb{R}^n)$. Although we recover the class of $K$-quasiregular mappings when $\sigma \equiv 0$, the theory of arbitrary solutions is significantly more complicated, partly due to the unavailability of a robust degree theory for non-quasiregular solutions. Nonetheless, we obtain a Liouville-type theorem and the sharp H\"older continuity estimate for all solutions, provided that $\sigma \in L^{n-\varepsilon}(\mathbb{R}^n) \cap L^{n+\varepsilon}(\mathbb{R}^n)$ for some $\varepsilon >0$. This gives an affirmative answer to a question of Astala, Iwaniec and Martin.