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Stable phase interfaces in the van der Waals--Cahn--Hilliard theory

We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is supported by an embedded smooth stable minimal hypersurface in low dimensions and an embedded smooth stable minimal hypersurface away from a closed singular set of co-dimension at least 7 in general dimensions. This result was previously known in case the critical points are local minimizers of energy, in which case the limit-hypersurface is locally area minimizing and its (normalized) multiplicity is 1 a.e. Our theorem uses earlier work of the first author establishing stability of the limit-interface as an integral varifold, and relies on a recent general theorem of the second author for its regularity conclusions in the presence of higher multiplicity.