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.
CITATION STYLE
Caroccia, M. (2017). Cheeger N-clusters. Calculus of Variations and Partial Differential Equations, 56(2). https://doi.org/10.1007/s00526-017-1109-9
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