Plateau's problem via the Allen--Cahn functional

Let $\Gamma$ be a compact codimension-two submanifold of $\mathbb{R}^n$, and let $L$ be a nontrivial real line bundle over $X = \mathbb{R}^n \setminus \Gamma$. We study the Allen--Cahn functional, $$E_\varepsilon(u) = \int_X \varepsilon \frac{|\nabla u|^2}{2} + \frac{(1-|u|^2)^2}{4\varepsilon}\,dx,$$ on the space of sections $u$ of $L$. Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to $\Gamma$. We first show that, for a family of critical sections with uniformly bounded energy, in the limit as $\varepsilon \to 0$, the associated family of energy measures converges to an integer rectifiable $(n-1)$-varifold $V$. Moreover, $V$ is stationary with respect to any variation which leaves $\Gamma$ fixed. Away from $\Gamma$, this follows from work of Hutchinson--Tonegawa; our result extends their interior theory up to the boundary $\Gamma$. Under additional hypotheses, we can say more about $V$. When $V$ arises as a limit of critical sections with uniformly bounded Morse index, $\Sigma := \operatorname{supp} \|V\|$ is a minimal hypersurface, smooth away from $\Gamma$ and a singular set of Hausdorff dimension at most $n-8$. If the sections are globally energy minimizing and $n = 3$, then $\Sigma$ is a smooth surface with boundary, $\partial \Sigma = \Gamma$ (at least if $L$ is chosen correctly), and $\Sigma$ has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fr\"{o}hlich and Struwe) that the smooth version of Plateau's problem admits a solution for every boundary curve in $\mathbb{R}^3$. This also works if $4 \leq n\leq 7$ and $\Gamma$ is assumed to lie in a strictly convex hypersurface.