On the behavior of 1-Laplacian ratio cuts on nearly rectangular domains

Abstract

Abstract The $p$-Laplacian has attracted more and more attention in data analysis disciplines in the past decade. However, there is still a knowledge gap about its behavior, which limits its practical application. In this paper, we are interested in its iterative behavior in domains contained in two-dimensional Euclidean space. Given a connected set $\varOmega _0 \subset \mathbb{R}^2$, define a sequence of sets $(\varOmega _n)_{n=0}^{\infty }$ where $\varOmega _{n+1}$ is the subset of $\varOmega _n$ where the first eigenfunction of the (properly normalized) Neumann $p$-Laplacian $ -\varDelta ^{(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 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 $\partial \varOmega _0$. 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 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.