Local minimizers in $\mathbb{R}^n$ of vector Allen-Cahn with an $(n+1)$-junction

Abstract

For a domain $\Omega$ that is a deformation of a unit ball in $\mathbb{R}^n$, we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy having $n+1$ wells. This sequence converges in the $L^1$ topology to a partition of $\Omega$ whose skeleton is given by a simplex cone that contains an $(n+1)$-junction 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. The results established in this article generalize those in the author's earlier article with Peter Sternberg (MR5033050), which dealt with the case $n=3$. We also weaken the one crucial assumption from the author's earlier article with Peter Sternberg (MR5033050).