2022/09/06 by Dong, Hongjie, Peng, Fa, Zhang, Yi Ru-Ya +1
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2209.02659
In dimension 2, we introduce a distributional Jacobian determinant det DVβ(Dv) for the nonlinear complex gradient (x1,x2)↦ |Dv|β(vx1,-vx2) for any β>-1, whenever v∈ W1,2 loc and β|Dv|1+β∈ W1,2loc. Then for any planar ∞-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 β=0, we develop an approach to build up a Liouville theorem, which improves that of Savin [33]. Precisely, if u is ∞-harmonic functions in whole \mathbb R2 with \liminfR→∞infc∈\mathbb R\frac1 R3∫B(0,R)|u(x)-c| dxlt;∞, then u=b+a⋅ x for some b∈\mathbb R and a∈\mathbb R2. (ii) Denoting by up the p-harmonic function having the same nonconstant boundary condition as u, we show that det DVβ(Dup) → det DVβ(Du) as p→∞ in the weak-⋆ sense in the space of Radon measure. Recall that Vβ(Dup) is always quasiregular mappings, but Vβ(Du) is not in general.