Classification of minimizing solutions to a two-dimensional Allen-Cahn system

Abstract: We study bounded entire solutions $u:\mathbb{R}^2\to \mathbb{R}<sup>2$ that minimize the Allen-Cahn functional \begin{equation*} J(u,Ω)=\int_Ω\left(\frac12 |\nabla u|^2+W(u)\right)\,d\mathbf{x}, \end{equation*} with the $D_3$-invariant triple-well potential \begin{equation*} W(u_1,u_2)=|u|^4+2u_1u_2^2-\frac23 u_1^3-|u|^2+\frac23. \end{equation*} 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 \begin{equation*} u_(\mathbf{x})=\sum_{i=1}^3 \frac{e^{\sqrt2 a_i\cdot \mathbf{x}}}{\sum_{j=1}^3 e^{\sqrt2 a_j\cdot \mathbf{x}}}a_i. \end{equation} We also demonstrate that the solutions obtained by minimizing within the $D_3$-equivariant class coincide with $u_*$. The key ingredient is a calibration identity arising from the special algebraic structure of $W$.