Some asymptotic results on the global shape of planar clusters

In this thesis we analyze the problem of the global shape of perimeter-minimizing planar N-clusters, which represent a model for soap bubbles, and of the creation of hexagonal patterns when the number of bubbles goes to infinity. We prove under some assumptions that the asymptotic global shape exists as a finite perimeter set, and we prove the mild regularity result that the set of points with zero Lebesgue density is an open set. Moreover the asymptotic perimeter density is that of a hexagonal lattice, up to lower order terms. We then consider some variants of this problem: -We consider an anisotropic perimeter with cubic Wulff shape. -We require all chambers to be squares (or hexagons). -We put different weights at different interfaces and prove a “sticky-disk” limit in a suitable regime. We then estimate the perimeter of the hexagonal honeycomb contained in a disk of radius r, proving that the error with respect to the natural area term is sublinear, and proving an analogue result for any periodic measure. Finally we consider the interface problem where we ask what is the optimal way to fill the region contained between two honeycombs whose orientations differ by a small angle, proving that some defects must appear and proving that, as the size of the chambers goes to zero, we can extract a “limit BV orientation” that describes the asymptotic orientation of the hexagonal chambers.