Remarks on the boundary set of spectral equipartitions

TL;DR

This paper investigates the geometric properties of spectral partitions, especially boundary length, in relation to minimal and equipartition sets on bounded domains and Riemannian manifolds.

Contribution

It explores whether properties of nodal sets extend to minimal spectral partitions and analyzes boundary set length in 2D spectral partitions.

Findings

Boundary set length bounds in 2D spectral partitions

Extension of nodal set properties to general spectral partitions

Insights into minimal partition boundary characteristics

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.