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

Dong, Hongjie; Peng, Fa; Zhang, Yi Ru-Ya; Zhou, Yuan

doi:10.1515/crelle-2024-0016

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 $v \in W_{loc}^{1, 2}$ v\in W^{1\smash{,}2}_{\mathrm{loc}}and $β {| D v |}^{1 + β} \in W_{loc}^{1, 2}$ \beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}}.This is new when $β \neq 0$ \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 $\underset{R \to \infty}{lim inf} inf_{c \in R} \frac{1}{R} ⨍_{B (0, R)} | u (x) - c | d x < \infty,$ \liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty,then $u = b + a \cdot x$ u=b+a\cdot xfor some $b \in R$ b\in\mathbb{R}and $a \in R^{2}$ 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 $det D V_{β} (D u_{p}) \to det D V_{β} (D u)$ \det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du)as $p \to \infty$ p\to\inftyin 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.

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