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

2025/02/24 by Abhishek Adimurthi, Adimurthi, Abhishek, Peter Sternberg +1
Computer Science · Materials Science · Mathematics · #35 #49 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.2502.17687

openalex publication_date 2025/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For Ω a perturbation of the unit ball in ℝ³, we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy. The sequence converges in L¹ 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.

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