关于谱极小划分 II：矩形情形

Spectral minimal partitions of unbounded metric graphs

We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form $-Δ+ V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying ``landscape'' on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum $Σ$ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any $k\in\…

谱最小分割的同调

Homology of spectral minimal partitions

本文通过定义修正拉普拉斯算子并借助代数拓扑中的相对同调工具，研究了非二分谱最小分割，证明其 Courant-sharp 特征函数的节点分割在特定拓扑类中最小，同时统一恢复了二分情形的已知结果。

A review on large k minimal spectral k-partitions and Pleijel's Theorem

In this survey, we review the properties of minimal spectral $k$-partitions in the two-dimensional case and revisit their connections with Pleijel's Theorem. We focus on the large $k$ problem (and the hexagonal conjecture) in connection with two recent preprints by J. Bourgain and S. Steinerberger on the Pleijel Theorem. This leads us also to discuss some conjecture by I. Polterovich, in relation with square tilings. We also establish a Pleijel Theorem for Aharonov-Bohm Hamiltonians and deduce from it, via the magnetic characterization of the minimal partitions, some lower bound for the number…

Examples of spectral minimal partitions

We study a minimal partition problem on the flat rectangular torus. We give a partial review of the existing literature, and present some numerical and theoretical work recently published elsewhere by V. Bonnaillie-No{ë}l and the author, with some improvements.

On a magnetic characterization of spectral minimal partitions

Given a bounded open set $Ω$ in $ \mathbb R^n$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_j$, we consider the quantity $\max_j λ(D_j)$ where $λ(D_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_j$. If we denote by $ \mathfrak L_k(Ω)$ the infimum over all the $k$-partitions of $ \max_j λ(D_j)$, a minimal $k$-partition is then a partition which realizes the infimum. When $k=2$, we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$'s is non trivial and quite interesting. In this paper, we give the p…

Asymptotics and estimates for spectral minimal partitions of metric graphs

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al, Calc. Var. 60 (2021), 61]. We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror the combinatorial structure of the metric graph as well. Combining them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic law similar to the well-known one…

组合图的谱极小划分

Spectral minimal partitions of combinatorial graphs

本文为加权图建立谱极小划分的存在性框架，引入典范紧化性与子图收敛概念，将无限图的谱极小问题转化为有限图研究，确保非紧情形下最优谱能量可实现。

Spectral minimal partitions for a family of tori

We study partitions of the rectangular two-dimensional flat torus of length 1 and width b into k domains, with b a parameter in (0, 1] and k an integer. We look for partitions which minimize the energy, definedas the largest first eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are inparticular interested in the way these minimal partitions change when b is varied. We present herean improvement, when k is odd, of the results on transition values of b established by B. Helffer andT. Hoffmann-Ostenhof (2014) and state a conjecture on those transition values. We establis…

度量图的Cheeger割与Robin谱极小划分

Cheeger cuts and Robin spectral minimal partitions of metric graphs

本文研究度量图上Cheeger型与Robin谱两类能量泛函的k划分问题，证明极小划分存在性，揭示Robin参数趋于0和无穷时分别收敛于k-Cheeger割与Dirichlet极小划分的渐近关系，并推测结果可推广到一般欧氏区域与流形。

无界区域的谱极小分划

Spectral minimal partitions of unbounded domains

本文研究无界区域的k-谱极小分划问题，通过阈值分析、集中紧性论证等方法，针对不同p值得出极小分划的存在性、基态性质及等分划特性等结论，并给出实例说明新现象。