On the behavior of $1$-Laplacian Ratio Cuts on nearly rectangular domains

TL;DR

This paper investigates the behavior of 1-Laplacian Ratio Cuts on nearly rectangular domains, showing convergence to rectangles with bounded eccentricity and analyzing the geometric evolution of domain shapes.

Contribution

It provides theoretical results on the convergence of Ratio Cuts to rectangles for domains close to rectangles, especially quadrilaterals, and discusses numerical aspects and open questions.

Findings

1-Laplacian spectral cut of near-rectangle domains is a flatter circular arc.

Quadrilaterals close to a rectangle with aspect ratio 2 tend to become more rectangular.

The study offers insights into domain shape evolution under Ratio Cut iterations.

Abstract

Given a connected set $\Omega_0 \subset \mathbb{R}^2$, define a sequence of sets $(\Omega_n)_{n=0}^{\infty}$ where $\Omega_{n+1}$ is the subset of $\Omega_n$ where the first eigenfunction of the (properly normalized) Neumann $p-$Laplacian $ -\Delta^{(p)} \phi = \lambda_1 |\phi|^{p-2} \phi$ is positive (or negative). For $p=1$, this is also referred to as the Ratio Cut of the domain. We conjecture that, unless $\Omega_0$ 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 $\partial \Omega_0$. 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.