Publication 16-CNA-017
Cheeger N-Clusters
Marco Caroccia
Dipartmento di Matematica
Università di Pisa
Largo Bruno Pontecorvo 5, 56127 Pisa,
Italy
caroccia.marco@gmail.com
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 Ω. 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 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.
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