On the heterogeneous distortion inequality

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 $x \mathbb{R}^n$. Here $K [1, \infty)$ is a constant and $\sigma \geq 0$ function in $L^n_{\mathrm{loc}}(\mathbb{R}^n)$. Although we recover class of $K$-quasiregular when \equiv 0$, theory arbitrary solutions significantly more complicated, partly due to unavailability robust degree non-quasiregular solutions. Nonetheless, obtain Liouville-type theorem sharp H\"older continuity estimate all solutions, provided L^{n-\varepsilon}(\mathbb{R}^n) \cap L^{n+\varepsilon}(\mathbb{R}^n)$ some $\varepsilon >0$. This gives an affirmative answer question Astala, Iwaniec Martin.