Cheeger $N$-clusters

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.

References in corpus (2)

• The Singular Structure and Regularity of Stationary and Minimizing Varifolds
• Improved convergence theorems for bubble clusters. I. The planar case

Cited by in corpus (6)

• On the higher Cheeger problem
• The Cheeger problem in abstract measure spaces
• The Cheeger N-problem in terms of BV functions
• On the first Robin eigenvalue of the Finsler $p$-Laplace operator as $p\to 1$
• Isoperimetric sets and $p$-Cheeger sets are in bijection
• The Pseudo-orthogonality for Graph $1$-Laplacian Eigenvectors and Applications to Higher Cheeger Constants and Data Clustering