  We study minimizers of the Allen-Cahn system. We consider the $ \varepsilon
$-energy functional with Dirichlet values and we establish the $ \Gamma
$-limit. The minimizers of the limiting functional are closely related to
minimizing partitions of the domain. Finally, utilizing that the triod and the
straight line are the only minimal cones in the plane together with regularity
results for minimal curves, we determine the precise structure of the
minimizers of the limiting functional, and thus the limit of minimizers of the
$ \varepsilon $-energy functional as $ \varepsilon \rightarrow 0 $.
