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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 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.
Mathematics Subject Classification49Q15
The author is grateful to professor Giovanni Alberti for his useful comments and for the useful discussions about this subject. Thanks to Gian Paolo Leonardi and Aldo Pratelli for having carefully read this work as a part of the Ph.D thesis of the author. The author is also grateful to Enea Parini for the useful comments about Proposition 4.4. The work of the author was partially supported by the project 2010A2TFX2-Calcolo delle Variazioni, funded by the Italian Ministry of Research and University. This work has been partially edited while the author was already a post-doc fellowship at the Carnegie Mellon University-Center for Nonlinear Analysis to which he is grateful.
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