On the isoperimetric problem with density for clusters

The aim of the thesis is to examine the generalization of the classical isoperimetric problem to the case of clusters with volume and perimeter weighted by densities, the latter possibly depending on the normal of the boundary of the measured set. The first two chapters are devoted to recalling the basic tools needed from Geometric Measure Theory, and to give an overview of the known literature on the isoperimetric problem with density for single sets. In the third chapter it is examined the case of converging densities at infinity. It is proved that any limit point of a minimizing sequence of clusters is still a minimizing cluster for its own volumes; moreover, a formula for the minimal perimeter is proved. The fourth chapter is focused on the ε − ε^β property and its consequences. This consists in the possibility to locally modify a set in order to slightly change its volume of a quantity ε keeping the variation of the perimeter controlled by a term C|ε|^β, with β related to the regularity of the perimeter density. The aim of the chapter is to generalize the property to clusters, with an explicit relation between the constant C and the local properties of the perimeter density. Boundedness and regularity of isoperimetric clusters are proved as applications. In the fifth chapter, some partial existence results for the isoperimetric problem with density for clusters are proved. In the sixth chapter, some open questions and future research projects are listed.