On the isoperimetric properties of Planar N-clusters

TL;DR

This thesis investigates isoperimetric properties of planar N-clusters, including energy distribution, stability of tilings, and generalized Cheeger constants, connecting to optimal partition problems and spectral theory.

Contribution

It provides new insights into the isoperimetric behavior of N-clusters, including stability results and a generalized Cheeger constant framework.

Findings

Equidistribution energy results for large N-clusters

Stability analysis of hexagonal tilings

Relation between generalized Cheeger constants and Laplacian eigenvalues

Abstract

This Thesis aims to highlight some isoperimetric questions involving the, so-called, $N$-clusters. We first briefly recall the theoretical framework we are adopting. This is done in Chapter one. In chapter two we focus on the standard isoperimetric problem for planar $N$-cluster for large values of $N$ and we provide an equidistribution energy-type results under some suitable assumption. The third Chapter is devoted to a stability results of the hexagonal honeycomb tiling. Finally in the fourth Chapter we consider a generalization of the Cheeger constant, defined as a minimization of a suitable energy among the class of the $N$-clusters. We show how this problem is related to the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian introduced by Caffarelli and Fang-Hua Lin in 2007. We conclude, in Chapter five, with some remarks and some possible future direction of investigation.