On the $\mathrm{\Gamma }$-convergence of the Allen-Cahn functional with boundary conditions

We study minimizers of the Allen-Cahn system. We consider the $\epsilon$-energy functional with Dirichlet values and we establish the $\mathrm{\Gamma }$-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 $\epsilon$-energy functional as $\epsilon \to 0$.