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

For Ω Ω a perturbation of the unit ball in R³ R 3 , we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy. The sequence converges in L¹ L 1 to a partition of Ω Ω 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 Γ Γ -limit of the sequence of Allen-Cahn functionals.