On the heterogeneous distortion inequality

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 \[\left|Df(x)\right|^n \leq K J_f(x) + σ^n(x) \left|f(x)\right|^n\] for almost every $x\in {\mathbb{R}}^n$. Here $K\in [1,\infty )$ is a constant and $\sigma \ge 0$ is a function in $L_{\mathrm{loc}}^n({\mathbb{R}}^n)$. 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 Hölder continuity estimate for all solutions, provided that $\sigma \in L^{n-\epsilon }({\mathbb{R}}^n)\cap L^{n+\epsilon }({\mathbb{R}}^n)$ for some $\epsilon >0$. This gives an affirmative answer to a question of Astala, Iwaniec and Martin.

Cited by in corpus (3)

• Mappings of generalized finite distortion and continuity
• A single-point Reshetnyak's theorem
• Fibers of monotone maps of finite distortion