论文标题
$σ_K$ -Loewner-Nirenberg问题的粘度解决方案的规律性
Regularity of viscosity solutions of the $σ_k$-Loewner-Nirenberg problem
论文作者
论文摘要
我们研究了$σ_K$ -LOEWNER-NIRENBERG问题的粘度解决方案的规律性$ u $在有限的平滑域$ω\ subset \ subset \ mathbb {r}^n $ for $ k \ geq 2 $上。众所周知，$ u $是$ω$的本地Lipschitz。我们证明，$ d $是$ \partialΩ$和$δ> 0 $的距离函数，$ u $在$ \ {0 <d（x）<δ\} $中是平滑的，第一个$（n-1）$衍生品$ d^{\ frac {\ frac {n-2） δ\} $。此外，我们确定了一个边界不变性，这是$ \ \partialΩ$及其协变量的主要曲率的多项式，并且仅当且仅当$ d^{\ frac {n-2} {2} {2}}} u $中时消失。利用环境流形的Schouten张量与几何测量理论的Submanifold和相关工具的平均曲率之间的关系，我们进一步证明，当$ \partialΩ$包含多个连接的组件时，$ u $在$ω$中无可区分。
We study the regularity of the viscosity solution $u$ of the $σ_k$-Loewner-Nirenberg problem on a bounded smooth domain $Ω\subset \mathbb{R}^n$ for $k \geq 2$. It was known that $u$ is locally Lipschitz in $Ω$. We prove that, with $d$ being the distance function to $\partialΩ$ and $δ> 0$ sufficiently small, $u$ is smooth in $\{0 < d(x) < δ\}$ and the first $(n-1)$ derivatives of $d^{\frac{n-2}{2}} u$ are Hölder continuous in $\{0 \leq d(x) < δ\}$. Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of $\partialΩ$ and its covariant derivatives and vanishes if and only if $d^{\frac{n-2}{2}} u$ is smooth in $\{0 \leq d(x) < δ\}$. Using a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when $\partialΩ$ contains more than one connected components, $u$ is not differentiable in $Ω$.
