2015/01/31 by Marco Caroccia
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Bounded function #Combinatorics #Conjecture #Discrete mathematics #Disjoint sets #Eigenvalues and eigenvectors #Geometric Analysis and Curvature Flows #Isoperimetric inequality #Laplace operator #Lebesgue integration #Lebesgue measure #Linear subspace #Mathematical analysis #Mathematics #Partition (number theory) #Pure mathematics #Spectral Theory in Mathematical Physics #math.AP #math.OC
paper · pdf · doi:10.1007/s00526-017-1109-9
arxiv created 2015/08/30 · openalex publication_date 2017/02/16 · arxiv updated 2017/03/31 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
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 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.