Spectral partition problems with volume and inclusion constraints

TL;DR

This paper investigates spectral partition problems with volume constraints, establishing existence, regularity, and qualitative properties of optimal partitions using variational and blowup techniques.

Contribution

It introduces a novel approach to spectral partitioning with measure constraints, proving existence and regularity of solutions through a weak formulation and advanced analytical methods.

Findings

Existence of optimal partitions with measure constraints

Eigenfunctions are locally Lipschitz continuous

Qualitative properties of the partitions are characterized

Abstract

In this paper, we discuss a class of spectral partition problems with a measure constraint, for partitions of a given bounded connected open set. We establish the existence of an optimal open partition, showing that the corresponding eigenfunctions are locally Lipschitz continuous, and obtain some qualitative properties for the partition. The proof uses an equivalent weak formulation that involves a minimization problem of a penalized functional where the variables are functions rather than domains, suitable deformations, blowup techniques, and a monotonicity formula.