Abstract For 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 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 -limit of the sequence of Allen-Cahn functionals.
Adimurthi et al. (Tue,) studied this question.