preprint

Inserted: 12 jun 2024

Year: 2022

Abstract:

In dimension 2, we introduce a distributional Jacobian determinant $\det DV_\beta(Dv)$ for the nonlinear complex gradient $(x_1,x_2)\mapsto
Dv
^\beta(v_{x_1},-v_{x_2})$ for any $\beta>-1$, whenever $v\in W^{1,2 }_{\text{loc}}$ and $\beta
Dv
^{1+\beta}\in W^{1,2}_{\text{loc}}$. Then for any planar $\infty$-harmonic function $u$, we show that such distributional Jacobian determinant is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications. (i) Applying this result with $\beta=0$, we develop an approach to build up a Liouville theorem, which improves that of Savin 33. Precisely, if $u$ is $\infty$-harmonic functions in whole ${\mathbb R}^2$ with $$ \liminf_{{R\to\infty}\inf{c\in\mathbb} R}\frac1 {R^3}\int_{B(0,R)}
u(x)-c
\,dx<\infty,$$ then $u=b+a\cdot x$ for some $b\in{\mathbb R}$ and $a\in{\mathbb R}^2$. (ii) Denoting by $u_p$ the $p$-harmonic function having the same nonconstant boundary condition as $u$, we show that $\det DV_\beta(Du_p) \to \det DV_\beta(Du)$ as $p\to\infty$ in the weak-$\star$ sense in the space of Radon measure. Recall that $V_\beta(Du_p)$ is always quasiregular mappings, but $V_\beta(Du)$ is not in general.