A note on non lower semicontinuous perimeter functionals on partitions

TL;DR

This paper investigates specific perimeter functionals on partitions in Euclidean space, identifying conditions for the existence and structure of minimizers, and analyzing their regularity.

Contribution

It introduces conditions ensuring minimizer existence for non lower semicontinuous perimeter functionals and characterizes their structure and regularity.

Findings

Established a domain condition for minimizer existence

Described the structure of minimizers

Analyzed regularity properties of minimizers

Abstract

We consider isotropic non lower semicontinuous weighted perimeter functionals defined on partitions of domains in $\mathbb{R}^n$. Besides identifying a condition on the structure of the domain which ensures the existence of minimizing configurations, we describe the structure of such minima, as well as their regularity.