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

Boundary behavior of limit-interfaces for the Allen-Cahn equation on Riemannian manifolds with Neumann boundary condition