*Phd Thesis*

**Inserted:** 22 jan 2016

**Last Updated:** 22 jan 2016

**Year:** 2015

**Notes:**

Advisors

Giovanni Alberti (Universita` di Pisa), Francesco Maggi (University of Texas at Austin).

**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.

**Download:**