In this paper we prove an asymptotic estimate, up to the second-order included, on the behaviour of the one-dimensional Allen–Cahn’s action functionals, around a periodic function with bounded variation and taking values in . The leading term of this estimate justifies and confirms, from a variational point of view, the results of Fusco–Hale [Dyn. Diff. Equation1 (1989), 75–94] and Carr–Pego [Comm. Pure Appl. Math.42 (1989), 523–576] on the exponentially slow motion of metastable patterns coexisting at the transition temperature.
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