Calculus of Variations and Geometric Measure Theory
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M. Caroccia

Cheeger $N$-clusters

created by caroccia on 22 Jan 2015
modified on 06 Jul 2017


Published Paper

Inserted: 22 jan 2015
Last Updated: 6 jul 2017

Journal: Calculus of variation and PDE's
Year: 2015


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.


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