2023/05/22 by Étienne Sandier, Sandier, Étienne, Peter Sternberg +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2305.13474
openalex publication_date 2023/05/22 · openalex created_date 2023/05/25 · openalex updated_date 2026/07/28
We construct an entire solution U:ℝ2→ℝ2 to the elliptic system ΔU=∇uW(U), where W:ℝ2→ [0,∞) is a `triple-well' potential. This solution is a local minimizer of the associated energy ∫ (1)/(2)|∇ U|2+W(U) dx in the sense that U minimizes the energy on any compact set among competitors agreeing with U outside that set. Furthermore, we show that along subsequences, the `blowdowns' of U given by UR(x):=U(Rx) approach a minimal triple junction as R→∞. Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions.