Asymptotic behavior of the free interface for entire vector minimizers in phase transitions

2021/12/01 by Alikakos, Nicholas D., Geng, Zhiyuan, Zarnescu, Arghir
#35B40 #35J47 #35J50 #35R35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.00796

Abstract

We study globally bounded entire minimizers u:ℝⁿ→ℝ^m of Allen-Cahn systems for potentials W≥ 0 with \W=0\=\a₁,...,a_N\ and W(u)∼ |u-a_i|^α near u=a_i, 00\ and the free boundary ∂ I₀. Furthermore, if α=1 we establish the upper bound H^n-1(∂^* I₀∩ B_r(x₀))≤ c₃r^n-1, r≥ r₀(x₀).

Related

• Entire minimizers of Allen-Cahn systems with sub-quadratic potentials
• Asymptotic limit of a vector-valued Allen-Cahn equation for phase transition dynamics
• Second order estimates for a free boundary phase transition
• Sharp lower bounds for vector Allen-Cahn energy and qualitative properties of minimizes under no symmetry hypotheses
• Boundary behavior of limit-interfaces for the Allen-Cahn equation on Riemannian manifolds with Neumann boundary condition