Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications

We introduce a distributional Jacobian determinant det D ⁢ V β ⁢ ( D ⁢ v ) \det DV_{\beta}(Dv) in dimension two for the nonlinear complex gradient V β ⁢ ( D ⁢ v ) = | D ⁢ v | β ⁢ ( v x 1 , − v x 2 ) V_{\beta}(Dv)=\lvert Dv\rvert^{\beta}(v_{x_{1}},-v_{x_{2}}) for any β > − 1 \beta>-1 , whenever v ∈ W loc 1 , 2 v\in W^{1\smash{,}2}_{\mathrm{loc}} and β ⁢ | D ⁢ v | 1 + β ∈ W loc 1 , 2 \beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}} . This is new when β ≠ 0 \beta\neq 0 . Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant det