Stability Criteria for Multiphase Partitioning Problems with Volume Constraints

TL;DR

This paper develops stability criteria for multiphase partitions with volume constraints in convex domains, focusing on two-phase contact scenarios and deriving conditions for stability based on eigenvalue analysis.

Contribution

It provides a detailed second variation formula and extends the Sternberg-Zumbrun instability result to multiple phases with volume constraints.

Findings

Derived stability criteria for multiphase partitions.

Recaptured and extended the Sternberg-Zumbrun instability result.

Analyzed the sign of the principal eigenvalue of the Jacobi operator.

Abstract

We study the stability of partitions involving two or more phases in convex domains under the assumption of at most two-phase contact, thus excluding in particular triple junctions. We present a detailed derivation of the second variation formula with particular attention to the boundary terms, and then study the sign of the principal eigenvalue of the Jacobi operator. We thus derive certain stability criteria, and in particular we recapture the Sternberg-Zumbrun result on the instability of the disconnected phases in the more general setting of several phases.