logo SBA

ETD

Archivio digitale delle tesi discusse presso l’Università di Pisa

Tesi etd-09192022-171221


Tipo di tesi
Tesi di dottorato di ricerca
Autore
SCATTAGLIA, VINCENZO
URN
etd-09192022-171221
Titolo
On the isoperimetric problem with density for clusters
Settore scientifico disciplinare
MAT/05
Corso di studi
MATEMATICA
Relatori
tutor Prof. Pratelli, Aldo
Parole chiave
  • calculus of variations
  • clusters
  • geometric measure theory
  • isoperimetric problems
  • partitioning problem
  • weighted measures
Data inizio appello
26/09/2022
Consultabilità
Completa
Riassunto
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.
File