Remarks on the boundary set of spectral equipartitions

Abstract

Given a bounded open set $\Omega$ in ${\mathbb{R}}^n$ (or a compact Riemannian manifold with boundary), and a partition of $\Omega$ by $k$ open sets ${\omega }_j$, we consider the quantity ${\max }_j\lambda ({\omega }_j)$, where $\lambda ({\omega }_j)$ is the ground state energy of the Dirichlet realization of the Laplacian in ${\omega }_j$. We denote by ${\mathfrak{L}}_k(\Omega )$ the infimum of ${\max }_j\lambda ({\omega }_j)$ over all $k$-partitions. A minimal $k$-partition is a partition which realizes the infimum. The purpose of this paper is to revisit properties of nodal sets and to explore if they are also true for minimal partitions, or more generally for spectral equipartitions. We focus on the length of the boundary set of the partition in the 2-dimensional situation.

arXiv commentFinal version to appear in the Philosophical Transactions of the Royal Society A

Cited by in corpus (4)

• Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle
• Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip
• Nodal domains, spectral minimal partitions, and their relation to Aharonov-Bohm operators
• An Euler-type formula for partitions of the Möbius strip