## 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 Ω_{0} ⊂ R^{2}, define a sequence of sets (Ω_{n})^{∞}_{n}_{=0} 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 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 ∂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 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.

Original language | English (US) |
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Pages (from-to) | 1563-1610 |

Number of pages | 48 |

Journal | Information and Inference |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2021 |

## Keywords

- domain deformation
- ratio cuts

## ASJC Scopus subject areas

- Analysis
- Statistics and Probability
- Numerical Analysis
- Computational Theory and Mathematics
- Applied Mathematics