Calendar

Posted January 12, 2026

Applied Analysis Seminar Questions or comments?

Daniel Massatt, New Jersey Institute of Technology
TBA

Event contact: Stephen Shipman

Posted January 11, 2026
Last modified March 6, 2026

Applied Analysis Seminar Questions or comments?

Zhiyuan Geng, Purdue University
Asymptotics for 2D vector-valued Allen-Cahn minimizers

For the scalar two-phase (elliptic) 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 two-dimensional 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. A key ingredient in our approach is the derivation of sharp upper and lower energy bounds via a slicing argument, which allows us to localize the diffuse interface within a small neighborhood of the sharp interface. As a consequence, we obtain a complete classification of global two-dimensional minimizers in terms of their blow-down limits at infinity. This is joint work with Nicholas Alikakos.

Posted March 6, 2026

Applied Analysis Seminar Questions or comments?

Yunfeng Zhang, University of Cincinnati
TBA

Event contact: Xiaoqi Huang