Local minimizers of the interface length functional based on a concept of local paired calibrations

TL;DR

This paper proves that regular flat partitions are locally minimizing for interface energy with respect to $L^1$ perturbations, using a localized paired calibration method applicable to a broad class of surface tension matrices.

Contribution

It introduces a localized paired calibration technique to establish local minimality of regular flat partitions for interface length functionals, extending to general surface tension matrices.

Findings

Regular flat partitions are locally minimizing for interface energy.

Stationary points of the length functional are necessarily regular flat partitions.

The method applies to a broad class of surface tension matrices.

Abstract

We establish that regular flat partitions are locally minimizing for the interface energy with respect to $L^1$ perturbations of the phases. Regular flat partitions are partitions of open sets in $\mathbb{R}^2$ whose network of interfaces consists of finitely many straight segments with a singular set made up of finitely many triple junctions at which the Herring angle condition is satisfied. This result not only holds for the case of the perimeter functional but for a general class of surface tension matrices. Our proof relies on a localized version of the paired calibration method which was introduced by Lawlor and Morgan (Pac. J. Appl. Math., 166(1), 1994) in conjunction with a relative energy functional that precisely captures the suboptimality of classical calibration estimates. Vice versa, we show that any stationary point of the length functional (in a sense of metric spaces) has to be a regular flat partition.