非局部$s$ -极小曲面与Lawson锥
Nonlocal $s$-minimal surfaces and Lawson cones
摘要：
非局部$s$ -$Σ= \partial E$的分数极小曲面方程，其中$E$是$R^N$中的开集，由$$ H_Σ^ s (p) := \int_{R^N} \frac {χ_E(x) - χ_{E^c}(x)} {|x-p|^{N+s}}\, dx \ =\ 0 \quad \text{for all } p\in Σ. $$给出。通过设$s\to 1$恢复了极小曲面的经典概念。在本文中，我们展示了第一个具体的例子（超越平面）的非局部$s-$极小曲面。当$s$接近于$1$时，我们首先构造一个连通的嵌入$s$-$R^3$中的极小旋转曲面，即{ \bf非局部悬链面}，标准悬链面$|x_3| = \log (r + \sqrt{r^2 -1})$的一个类似物。而不是最终的对数增长，这个表面变得渐近锥$|x_3|= r\sqrt{1-s}$。我们还发现了一个两片嵌入$s$ -最小表面渐近相同的锥，一个类似的简单工会的两个平行的平面。另一方面，对于任何$0<s<1$，$n,m\ge 1$，$s-$最小劳森锥$|v|=α|u|$，$(u,v)\in R^n\times R^m$，被发现存在。在与经典的情况下形成鲜明对比，我们证明了他们的稳定性小$s$和$n+m=7$，这表明，与经典理论（或$s$接近1的情况下），$s$面积最小化表面的规律可能不成立的尺寸$7$。
英文摘要：
The nonlocal $s$-fractional minimal surface equation for $Σ= \partial E$ where $E$ is an open set in $R^N$ is given by $$ H_Σ^ s (p) := \int_{R^N} \frac {χ_E(x) - χ_{E^c}(x)} {|x-p|^{N+s}}\, dx \ =\ 0 \quad \text{for all } p\in Σ. $$ Here $0<s<1$, $χ$ designates characteristic function, and the integral is understood in the principal value sense. The classical notion of minimal surface is recovered by letting $s\to 1$. In this paper we exhibit the first concrete examples (beyond the plane) of nonlocal $s-$minimal surfaces. When $s$ is close to $1$, we first construct a connected embedded $s$-minimal surface of revolution in $R^3$, the {\bf nonlocal catenoid}, an analog of the standard catenoid $|x_3| = \log (r + \sqrt{r^2 -1})$. Rather than eventual logarithmic growth, this surface becomes asymptotic to the cone $|x_3|= r\sqrt{1-s}$. We also find a two-sheet embedded $s$-minimal surface asymptotic to the same cone, an analog to the simple union of two parallel planes. On the other hand, for any $0<s<1$, $n,m\ge 1$, $s-$minimal Lawson cones $|v|=α|u|$, $(u,v)\in R^n\times R^m$, are found to exist. In sharp contrast with the classical case, we prove their stability for small $s$ and $n+m=7$, which suggests that unlike the classical theory (or the case $s$ close to 1), the regularity of $s$-area minimizing surfaces may not hold true in dimension $7$.
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