Cheeger $N$-clusters

TL;DR

This paper introduces a Cheeger-type constant for N-clusters in bounded domains, analyzes their regularity and structure, and explores their connection to the Laplacian eigenvalue partition problem and related conjectures.

Contribution

It defines a new Cheeger N-cluster constant, studies their regularity and structure, and links this to the eigenvalue partition problem and Caffarelli-Lin's conjecture.

Findings

Provides a lower bound for the first Dirichlet eigenvalue partition problem.

Describes the structure of Cheeger N-clusters in the plane.

Establishes relations between the Cheeger constant and spectral partitioning.

Abstract

In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the $N$-clusters contained in an open bounded set $\Omega$. Here with $N$-Cluster we mean a family of $N$ sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any $N$-cluster attaining such a minimum a Cheeger $N$-cluster. Our purpose is to provide a non trivial lower bound on the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. Here we discuss the regularity of Cheeger $N$-clusters in a general ambient space dimension and we give a precise description of their structure in the the planar case. The last part is devoted to the relation between the functional introduced here (namely the $N$-Cheeger constant), the partition problem for the first Dirichlet eigenvalue of the Laplacian and the Caffarelli and Lin's conjecture.