Plateau's problem via the Allen--Cahn functional

Let $\mathrm{\Gamma }$ be a compact codimension-two submanifold of ${\mathbb{R}}^n$, and let $L$ be a nontrivial real line bundle over $X={\mathbb{R}}^n\setminus \mathrm{\Gamma }$. We study the Allen–Cahn functional, $$E_{\epsilon }(u)={\int }_X\epsilon \frac{\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2}{2}+\frac{(1-\mathrm{\mid }u{\mathrm{\mid }}^2{)}^2}{4\epsilon }dx,$$ 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 $\mathrm{\Gamma }$. We first show that, for a family of critical sections with uniformly bounded energy, in the limit as $\epsilon \to 0$, 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 $\mathrm{\Gamma }$ fixed. Away from $\mathrm{\Gamma }$, this follows from work of Hutchinson–Tonegawa; our result extends their interior theory up to the boundary $\mathrm{\Gamma }$. Under additional hypotheses, we can say more about $V$. When $V$ arises as a limit of critical sections with uniformly bounded Morse index, $\mathrm{\Sigma }:=\mathrm{supp}\mathrm{\parallel }V\mathrm{\parallel }$ is a minimal hypersurface, smooth away from $\mathrm{\Gamma }$ and a singular set of Hausdorff dimension at most $n-8$. If the sections are globally energy minimizing and $n=3$, then $\mathrm{\Sigma }$ is a smooth surface with boundary, $\mathrm{\partial }\mathrm{\Sigma }=\mathrm{\Gamma }$ (at least if $L$ is chosen correctly), and $\mathrm{\Sigma }$ has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich 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 $4\le n\le 7$ and $\mathrm{\Gamma }$ is assumed to lie in a strictly convex hypersurface.