We introduce a distributional Jacobian determinant \det D V_{\beta}(Dv) in dimension two for the nonlinear complex gradient V_{\beta}(Dv) = |Dv|^{\beta}(vx_1, -vx_2) for any \beta > -1, whenever v \in W^{1,2}_{\text{loc}} and \beta |Dv|^{1+\beta} \in W^{1,2}_{\text{loc}}. This is new when \beta \neq 0. Given any planar \infty-harmonic function u, we show that such distributional Jacobian determinant \det D V_{\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 \beta = 0, we develop an approach to build up a Liouville theorem, which improves that of Savin. Precisely, if u is an \infty-harmonic function in the whole \mathbb{R}^2 with \liminf_{R \to \infty} \inf_{c \in \mathbb{R}} \frac{1}{R} \int_{B(0,R)} |u(x) - c| dx < \infty, then u = b + a \cdot x for some b \in \mathbb{R} and a \in \mathbb{R}^2. Denoting by u_p the p-harmonic function having the same nonconstant boundary condition as u, we show that \det D V_{\beta}(Du_p) \to \det D V_{\beta}(Du) as p \to \infty in the weak-\ast 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.

Publication:

Journal Fur Die Reine Und Angewandte Mathematik (2024)

http://dx.doi.org/10.1515/crelle-2024-0016

Author:

Hongjie Dong

Division of Applied Mathematics, 6752 Brown University, Providence RI 02912, USA

Fa Peng

School of Mathematical Sciences, Beihang University , Beijing, P. R. China

Yi Ru-Ya Zhang

Academy of Mathematics and Systems Science, The Chinese Academy of Sciences , Beijing, P. R. China

Email: yzhang@amss.ac.cn

Yuan Zhou

School of Mathematical Sciences, Beijing Normal University, Haidian District Xinejikou Waidajie No. 19, Beijing, P. R. China