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

Abstract

Abstract 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 D ⁢ V β ⁢ ( D ⁢ u ) \det DV_{\beta}(Du) 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 \beta=0 , we develop an approach to build up a Liouville theorem, which improves that of Savin. Precisely, if 𝑢 is an ∞-harmonic function in the whole R 2 \mathbb{R}^{2} with lim inf R → ∞ inf c ∈ R 1 R ⁢ ⨍ B ⁢ ( 0 , R ) | u ⁢ ( x ) − c | ⁢ d x < ∞ , \liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty, then u = b + a ⋅ x u=b+a\cdot x for some b ∈ R b\in\mathbb{R} and a ∈ R 2 a\in\mathbb{R}^{2} . Denoting by u p u_{p} the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that det D ⁢ V β ⁢ ( D ⁢ u p ) → det D ⁢ V β ⁢ ( D ⁢ u ) \det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du) as p → ∞ p\to\infty in the weak-⋆ sense in the space of Radon measure. Recall that V β ⁢ ( D ⁢ u p ) V_{\beta}(Du_{p}) is always quasiregular mappings, but V β ⁢ ( D ⁢ u ) V_{\beta}(Du) is not in general.