Asymptotics for 2D vector-valued Allen-Cahn minimizers
数学专题报告
报告题目(Title)：Asymptotics for 2D vector-valued Allen-Cahn minimizers
报告人(Speaker)：Zhiyuan Geng (Purdue University)
地点(Place)：ZOOM ID：985 9910 8529 PWD：534602
时间(Time)：2026年3月6日（周五）9:30-10:30
邀请人(Inviter)：熊金钢
报告摘要
For the scalar two-phase Allen-Cahn equation, there is a rich literature on the celebrated De Giorgi conjecture, which reveals deep connections between diffuse interfaces and minimal surfaces. On the other hand, for three or more equally preferred phases, a vector-valued order parameter is required, and the resulting diffuse interfaces are expected to resemble weighted minimal partitions. In this talk, I will present recent results on minimizers of a 2D Allen-Cahn system with a multi-well potential. We describe the asymptotic behavior near the junction of three phases by analyzing the blow-up limit, which is a global minimizing solution converging at infinity to a Y-shaped minimal cone. We derive sharp energy upper and lower bounds via a novel slicing argument, which allows us to localize the diffuse interface within a small neighborhood of the sharp interface. In addition, we obtain a classification of global 2D minimizers in terms of their blow-down limits at infinity. This is joint work with Nicholas Alikakos.
* This PDE seminar is co-organized with Tianling Jin at HKUST and Juncheng Wei at CUHK. See the seminar webpage: https://www.math.hkust.edu.hk/~tianlingjin/PDEseminar.html
