Boundary Behavior of Limit-Interfaces for the Allen–Cahn Equation on Riemannian Manifolds with Neumann Boundary Condition

We study the boundary behavior of any limit-interface arising from a sequence of general critical points of the Allen–Cahn energy functionals on a smooth bounded domain. Given any such sequence with uniform energy bounds, we prove that the limit-interface is a free boundary varifold which is integer rectifiable up to the boundary. This extends earlier work of Hutchinson and Tonegawa on the interior regularity of the limit-interface. A key novelty in our result is that no convexity assumption of the boundary is required and it is valid even when the limit-interface clusters near the boundary. Moreover, our arguments are local and thus work in the Riemannian setting. This work provides the first step towards the regularity theory for the Allen–Cahn min-max theory for free boundary minimal hypersurfaces, which was developed in the Almgren–Pitts setting by the first-named author and Zhou.