论文标题
（非线性）平面$ \ infty $ harmonic函数和应用的jacobian决定因素
Jacobian determinants for (nonlinear) gradient of planar $\infty$-harmonic functions and applications
论文作者
论文摘要
在维度2中，我们引入了一个分布分布的jacobian jacobian $ \ detdv_β（dv）$，用于非线性复杂梯度$（x_1，x_1，x_2）\ mapsto | dv |^β（v_ {x_1}， - v_ {x_2}}， - v_ {x_2}）$ } _ {\ text {loc}} $和$β| dv |^{1+β} \ in W^{1,2} _ {\ text {loc}} $。然后，对于任何平面$ \ infty $ harmonic函数$ u $，我们表明这种分销雅各布决定因素是一种非负ra量，具有一些定量的局部下限和上限。我们还提供以下两个应用程序。 （i）使用$β= 0 $应用此结果，我们开发了一种构建liouville定理的方法，该方法改善了萨维尔[33]。确切地说，如果$ u $是$ \ infty $ -Harmonic函数，则全$ {\ mathbb r}^2 $ with $ \ liminf_ {r \ to \ infty} \ infty} \ inf_ {c \ in \ mathbb r} \ frac1 \ frac1 {r^3} \ int_ {b（0，r）} | u（x）-c | \，dx <\ infty，$$，然后$ u = b+a \ cdot x $用于某些$ b \ in {\ mathbb r} $ in {\ mathbb r} $和$ a \ in {\ mathbb r}^2 $。 （ii）用$ u_p $表示$ p $ - 谐波函数具有与$ u $相同的非稳定边界条件，我们表明$ \ detdv_β（du_p）\ to \ detdv_β（du）dv_β（du）$ as $ p \ to $ p \ to \ to \ to \ to \ infty to f to f the弱-$ \ $ \ $ \ $ \ $ \ $ \ $ \ Star $在Radon space of Radon Matues of Radon sense of Radon。回想一下，$v_β（du_p）$始终是Quasiregular映射，但是$v_β（du）$通常不是。
In dimension 2, we introduce a distributional Jacobian determinant $\det DV_β(Dv)$ for the nonlinear complex gradient $(x_1,x_2)\mapsto |Dv|^β(v_{x_1},-v_{x_2})$ for any $β>-1$, whenever $v\in W^{1,2 }_{\text{loc}}$ and $β|Dv|^{1+β}\in W^{1,2}_{\text{loc}}$. Then for any planar $\infty$-harmonic function $u$, we show that such distributional Jacobian determinant is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications. (i) Applying this result with $β=0$, we develop an approach to build up a Liouville theorem, which improves that of Savin [33]. Precisely, if $u$ is $\infty$-harmonic functions in whole ${\mathbb R}^2$ with $$ \liminf_{R\to\infty}\inf_{c\in\mathbb R}\frac1 {R^3}\int_{B(0,R)}|u(x)-c|\,dx<\infty,$$ then $u=b+a\cdot x$ for some $b\in{\mathbb R}$ and $a\in{\mathbb R}^2$. (ii) Denoting by $u_p$ the $p$-harmonic function having the same nonconstant boundary condition as $u$, we show that $\det DV_β(Du_p) \to \det DV_β(Du)$ as $p\to\infty$ in the weak-$\star$ sense in the space of Radon measure. Recall that $V_β(Du_p)$ is always quasiregular mappings, but $V_β(Du)$ is not in general.
