近似矩形域上$1$ -Laplacian比割的性质
On the behavior of $1$-Laplacian Ratio Cuts on nearly rectangular domains
摘要：
给定一个连通集$Ω_0 \subset \mathbb{R}^2$，定义一个集合序列$(Ω_n)_{n=0}^{\infty}$，其中$Ω_{n+1}$是$Ω_n$的子集，其中（正确归一化的）Neumann $p-$ Laplacian $ -Δ^{(p)} ϕ= λ_1 |ϕ|^{p-2} ϕ$的第一本征函数为正（或负）。对于$p=1$，这也称为结构域的比率切割。我们猜想，除非$Ω_0$是一个等腰直角三角形，这些集合收敛到一组矩形的偏心率由2界定的Gromov-Hausdorff距离，只要他们有一定的距离边界$\partial Ω_0$。我们建立了$p=1$的这个猜想的一些方面，其中我们证明了：（1）充分接近给定长宽比的矩形的区域的1-Laplacian谱割是比原始区域更接近平坦的圆弧（最终导致四分之一）和（2）接近于纵横比$2$的矩形的四边形保持接近四边形，并且以适当的度量移动接近于矩形。我们还讨论了一些数值方面，并提出了许多悬而未决的问题。
英文摘要：
Given a connected set $Ω_0 \subset \mathbb{R}^2$, define a sequence of sets $(Ω_n)_{n=0}^{\infty}$ where $Ω_{n+1}$ is the subset of $Ω_n$ where the first eigenfunction of the (properly normalized) Neumann $p-$Laplacian $ -Δ^{(p)} ϕ= λ_1 |ϕ|^{p-2} ϕ$ is positive (or negative). For $p=1$, this is also referred to as the Ratio Cut of the domain. We conjecture that, unless $Ω_0$ is an isosceles right triangle, these sets converge to the set of rectangles with eccentricity bounded by 2 in the Gromov-Hausdorff distance as long as they have a certain distance to the boundary $\partial Ω_0$. We establish some aspects of this conjecture for $p=1$ where we prove that (1) the 1-Laplacian spectral cut of domains sufficiently close to rectangles of a given aspect ratio is a circular arc that is closer to flat than the original domain (leading eventually to quadrilaterals) and (2) quadrilaterals close to a rectangle of aspect ratio $2$ stay close to quadrilaterals and move closer to rectangles in a suitable metric. We also discuss some numerical aspects and pose many open questions.
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