2023/10/04 by James B. Kennedy, Kennedy, James B., João P. Ribeiro +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Graph theory and applications #Nonlinear Partial Differential Equations #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2310.02701
openalex publication_date 2023/10/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study partition problems based on two ostensibly different kinds of energy functionals defined on k-partitions of metric graphs: Cheeger-type functionals whose minimisers are the k-Cheeger cuts of the graph, and the corresponding values are the k-Cheeger constants of the graph; and functionals built using the first eigenvalue of the Laplacian with positive, i.e. absorbing, Robin (delta) vertex conditions at the boundary of the partition elements. We prove existence of minimising k-partitions, k ≥ 2, for both these functionals. We also show that, for each k ≥ 2, as the Robin parameter α→ 0, up to a renormalisation the spectral minimal Robin energy converges to the k-Cheeger constant. Moreover, up to a subsequence, the Robin spectral minimal k-partitions converge in a natural sense to a k-Cheeger cut of the graph. Finally, we show that as α→ ∞ there is convergence in a similar sense to the corresponding Dirichlet minimal energy and partitions. It is strongly expected that similar results hold on general (smooth, bounded) Euclidean domains and manifolds.