Cheeger $N$-clusters
Abstract
In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the -clusters contained in an open bounded set . Here with -Cluster we mean a family of sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any -cluster attaining such a minimum a Cheeger -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 -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 -Cheeger constant), the partition problem for the first Dirichlet eigenvalue of the Laplacian and the Caffarelli and Lin's conjecture.
Cite
@article{arxiv.1501.05923,
title = {Cheeger $N$-clusters},
author = {Marco Caroccia},
journal= {arXiv preprint arXiv:1501.05923},
year = {2017}
}