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

$$\begin{aligned} \left| Df(x) \right| ^n \le K J_f(x) + \sigma ^n(x) \left| f(x) \right| ^n \end{aligned}$$

for almost every $x \in \mathbb {R}^n$. Here $K \in [1, \infty )$ is a constant and $\sigma \ge 0$ is a function in $L^n_\mathrm {loc}(\mathbb {R}^n)$. 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 Hölder 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.