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

We introduce a distributional Jacobian determinant det D ⁢ V β ⁢ (D ⁢ v) in dimension two for the nonlinear complex gradient V β ⁢ (D ⁢ v) = | D ⁢ v | β ⁢ (v x 1 , − v x 2 ) for any β > − 1 , whenever v ∈ W loc 1 , 2 and β ⁢ | D ⁢ v | 1 + β ∈ W loc 1 , 2 . This is new when β ≠ 0 . Given any planar ∞-harmonic function 푢, we show that such distributional Jacobian determinant det D ⁢ V β ⁢ (D ⁢ u) is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications. Applying this result with β = 0 , we develop an approach to build up a Liouville theorem, which improves that of Savin. Precisely, if 푢 is an ∞-harmonic function in the whole R 2 with lim inf R → ∞ inf c ∈ R 1 R ⁢ ⨍ B ⁢ (0 , R) | u ⁢ (x) − c | ⁢ d x < ∞ , then u = b + a ⋅ x for some b ∈ R and a ∈ R 2 . Denoting by u p the 푝-harmonic function having the same nonconstant boundary condition as 푢, we show that det D ⁢ V β ⁢ (D ⁢ u p) → det D ⁢ V β ⁢ (D ⁢ u) as p → ∞ in the weak-⋆ sense in the space of Radon measure. Recall that V β ⁢ (D ⁢ u p) is always quasiregular mappings, but V β ⁢ (D ⁢ u) is not in general. [ABSTRACT FROM AUTHOR]