Plateau’s problem via the Allen–Cahn functional

Let Γ Γ be a compact codimension-two submanifold of Rⁿ R n , and let L be a nontrivial real line bundle over X = Rⁿ Γ X = R n \ Γ . We study the Allen–Cahn functional, alignedE_ε (u) = ∫ X ε |∇ u|²/2 + (1-|u|²)²/4ε \,dx, \ E ε ( u ) = ∫ X ε | ∇ u | 2 2 + ( 1 - | u | 2 ) 2 4 ε d x , 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 Γ Γ . We first show that, for a family of critical sections with uniformly bounded energy, in the limit as ε → 0 ε → 0 , the associated family of energy measures converges to an integer rectifiable $$(n-1)$$ ( n - 1 ) -varifold V . Moreover, V is stationary with respect to any variation which leaves Γ Γ fixed. Away from Γ Γ , this follows from work of Hutchinson–Tonegawa; our result extends their interior theory up to the boundary Γ Γ . Under additional hypotheses, we can say more about V . When V arises as a limit of critical sections with uniformly bounded Morse index, Σ := \,supp\, V Σ : = supp ‖ V ‖ is a minimal hypersurface, smooth away from Γ Γ and a singular set of Hausdorff dimension at most $$n-8$$ n - 8 . If the sections are globally energy minimizing and $$n = 3$$ n = 3 , then Σ Σ is a smooth surface with boundary, ∂ Σ = Γ ∂ Σ = Γ (at least if L is chosen correctly), and Σ Σ 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 R³ R 3 . This also works if 4 ≤ n≤ 7 4 ≤ n ≤ 7 and Γ Γ is assumed to lie in a strictly convex hypersurface.