Jacobian determinants for nonlinear gradient of planar $\infty$-harmonic functions and applications

In dimension 2, we introduce a distributional Jacobian determinant for the nonlinear complex gradient $V_\beta(Dv)$ of a function $v\in W^{1,2 }_{\mathrm{loc}}$ with $\beta |Dv|^{1+\beta}\in W^{1,2}_{\mathrm{loc}}$, where $\beta>-1$. This is new when $\beta\ne0$. Given any planar $\infty$-harmonic function $u$, we show that such distributional Jacobian determinant $\det DV_\beta(Du)$ is a nonnegative Radon measure with some quantitative local lower and upper bounds. 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.