On the continuity of functionals defined on partitions

TL;DR

This paper characterizes the conditions under which certain functionals on finite partitions are continuous, showing that in the multiphase case, continuity is tied to the convergence of the perimeter of the jump set, extending classical measure theory results.

Contribution

It extends the classical Reshetnyak continuity theorem to multiphase partitions, establishing a new criterion for functional continuity based on perimeter convergence.

Findings

Continuity of functionals is equivalent to perimeter convergence in multiphase cases.

Provides a characterization of functional continuity on finite Caccioppoli partitions.

Extends classical measure theory results to multiphase partition settings.

Abstract

We characterize the continuity of prototypical functionals acting on finite Caccioppoli partitions. In the spirit of the classical Reshetnyak continuity theorem for measures that can be used to prove continuity of surface-type functionals defined on single sets of finite perimeter we show that in the multiphase case continuity is equivalent to convergence of the perimeter of the jump set.