2026/07/22 by Abhishek Adimurthi
#math.AP #math.OC
For a domain Ω that is a deformation of a unit ball in ℝn, 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 L1 topology to a partition of Ω 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 Γ-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).