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 Ω0 ⊂ R2, define a sequence of sets (Ωn)∞n=0 where Ωn+1 is the subset of Ωn where the first eigenfunction of the (properly normalized) Neumann p-Laplacian −Δ(p)φ = λ1|φ|p−2φ 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 ∂Ω0. 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