On The Phase Connectedness Of The Volume-Constrained Area Minimizing Partitioning Problem

TL;DR

This paper investigates the stability of multi-phase partitions in convex domains, deriving second variation formulas and demonstrating the existence of stable configurations with disconnected phases, extending previous two-phase results.

Contribution

It provides a detailed derivation of second variation formulas for three-phase partitions and proves the existence of stable, disconnected phase configurations, advancing understanding of multi-phase area minimization.

Findings

Derived second variation formula with boundary and spine terms

Proved existence of stable partitions with disconnected phases

Extended stability analysis from two-phase to three-phase cases

Abstract

We study the stability of partitions in convex domains involving simultaneous coexistence of three phases, viz. triple junctions. We present a careful derivation of the formula for the second variation of area, written in a suitable form with particular attention to boundary and spine terms, and prove, in contrast to the two phase case, the existence of stable partitions involving a disconnected phase.