2026/07/22 by Zhiyuan Geng
#math.AP
We study bounded entire solutions u:ℝ2→ ℝ2 that minimize the Allen-Cahn functional J(u,Ω)=∫Ω(\frac12 |∇ u|2+W(u)) dx, with the D3-invariant triple-well potential W(u1,u2)=|u|4+2u1u22-\frac23 u13-|u|2+\frac23. We obtain a complete classification of entire minimizing solutions. In particular, when u has a triple-junction structure at infinity, up to translation and orthogonal change of coordinates, u has the explicit profile u_*(x)=∑i=13 \frace√2 ai⋅ x∑j=13 e√2 aj⋅ xai. We also demonstrate that the solutions obtained by minimizing within the D3-equivariant class coincide with u_*. The key ingredient is a calibration identity arising from the special algebraic structure of W.