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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We study the Γ-convergence of damage to fracture energy functionals in the presence of low-order nonlinear potentials that allows us to model physical phenomena such as fluid-driven fracturing, plastic slip, and the satisfaction of kinematical constraints such as crack non-interpenetration. Existence results are also addressed.

Abstract. We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics.

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We prove that quadratic pair interactions for functions defined on planar Poisson clouds and taking into account pairs of sites of distance up to a certain (large-enough) threshold can be almost surely approximated by the multiple of the Dirichlet energy by a deterministic constant. This is achieved by scaling the Poisson cloud and the corresponding energies and computing a compact discrete-to-continuum limit. In order to avoid the effect of exceptional regions of the Poisson cloud, with an accumulation of sites or with 'disconnected' sites, a suitable 'coarse-grained' notion of convergence of functions defined on scaled Poisson clouds must be given. Contents1. Introduction 2. Notation and preliminaries 2.1. General notation 2.2. Poisson point clouds 2.3. Dirichlet energy on point clouds 2.4. Voronoi cells and paths 2.5. Piecewise-constant extensions 2.6. Geometric structure of Poisson point processes 2.7. A notion of convergence for functions on Poisson point clouds 3. The main results 4. Proof of the compactness theorem 4.1. Preliminary lemmas 4.2. Properties of the convergence for sequences with equibounded energy 4.3. Proof of the compactness theorem 5. Proof of the Γ-convergence Theorem 3.3 5.1. The cell problem 5.2. A boundary-value fixing argument 5.3. Proof of the lower bound 5.4. Proof of the upper bound 6. Appendix 6.1. Existence of regular grids

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