Local minimizers in $3$d of vector Allen-Cahn with a quadruple junction

TL;DR

This paper proves the existence of local minimizers for the vector Allen-Cahn energy in three dimensions, converging to a partition with a quadruple junction, using $ ext{Gamma}$-convergence and perimeter minimization techniques.

Contribution

It demonstrates the existence of local minimizers with quadruple junctions in 3D vector Allen-Cahn models, a novel geometric configuration not previously established.

Findings

Existence of local minimizers converging to tetrahedral partitions

Identification of quadruple junctions as local minimizers

Application of $ ext{Gamma}$-convergence to analyze minimizers

Abstract

For $\Omega$ a perturbation of the unit ball in $\mathbb{R}^3$, we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy. The sequence converges in $L^1$ to a partition of $\Omega$ whose skeleton is given by a tetrahedral cone and thus contains a quadruple point. This is accomplished by proving that the partition is an isolated local minimizer of a weighted perimeter problem arising as the associated $\Gamma$-limit of the sequence of Allen-Cahn functionals.