Institut Fourier, Université Grenoble 1 and CNRS, , BP74, 38 402 St Martin d'Hères Cedex, France.
Philos Trans A Math Phys Eng Sci. 2013 Dec 16;372(2007):20120492. doi: 10.1098/rsta.2012.0492. Print 2014 Jan 28.
Given a bounded open set in R2 (or in a Riemannian manifold), and a partition of Ω by k open sets ωj, we consider the quantity maxj λ(ωj), where λ(ωj) is the ground state energy of the Dirichlet realization of the Laplacian in ωj. We denote by Lk(Ω) the infimum of maxj λ(ω) over all k-partitions. A minimal k-partition is a partition that realizes the infimum. Although the analysis of minimal k-partitions is rather standard when k=2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular spectral equipartitions, i.e. the ground state energies λ(ωj) are all equal. The purpose of this paper is to revisit various properties of nodal sets, and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We prove a lower bound for the length of the boundary set of a partition in the two-dimensional situation. We consider estimates involving the cardinality of the partition.
给定 R2 中的有界开集（或黎曼流形），以及 Ω 通过 k 个开集 ωj 的分区，我们考虑数量 maxj λ(ωj)，其中 λ(ωj) 是 ωj 中拉普拉斯算子的狄利克雷实现的基态能量。我们将 Lk(Ω) 定义为所有 k-分区的 maxj λ(ω) 的下确界。最小 k-分区是实现该下确界的分区。尽管当 k=2 时，最小 k-分区的分析相当标准（我们找到第二个特征函数的节点域），但对于更高值的 k，分析变得非平凡且非常有趣。最小分区特别是谱等分，即基态能量 λ(ωj) 都相等。本文的目的是重新研究节点集的各种性质，并探讨它们是否也适用于最小分区，或者更一般地适用于谱等分。我们证明了二维情况下分区边界集长度的下界。我们考虑了涉及分区基数的估计。
