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

We study bounded entire solutions $u:{\mathbb{R}}^2\to {\mathbb{R}}^2$ that minimize the Allen-Cahn functional \begin{equation*} J(u,\Omega)=\int_\Omega \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 $u_{\ast }$. The key ingredient is a calibration identity arising from the special algebraic structure of $W$.