Abstract
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 {±1}. The leading term of this estimate justifies and confirms, from a variational point of view, the results of Fusco-Hale (Dyn. Diff. Equation 1 (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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