Sharp-interface limit of the Allen-Cahn action functional in one space dimension

Robert V. Kohn, Maria G. Reznikoff, Yoshihiro Tonegawa

Research output: Contribution to journalArticlepeer-review


We analyze the sharp-interface limit of the action minimization problem for the stochastically perturbed Allen-Cahn equation in one space dimension. The action is a deterministic functional which is linked to the behavior of the stochastic process in the small noise limit. Previously, heuristic arguments and numerical results have suggested that the limiting action should "count" two competing costs: the cost to nucleate interfaces and the cost to propagate them. In addition, constructions have been used to derive an upper bound for the minimal action which was proved optimal on the level of scaling. In this paper, we prove that for d = 1, the upper bound achieved by the constructions is in fact sharp. Furthermore, we derive a lower bound for the functional itself, which is in agreement with the heuristic picture. To do so, we characterize the sharp-interface limit of the space-time energy measures. The proof relies on an extension of earlier results for the related elliptic problem.

Original languageEnglish (US)
Pages (from-to)503-534
Number of pages32
JournalCalculus of Variations and Partial Differential Equations
Issue number4
StatePublished - Apr 2006


  • Action minimization
  • Allen-Cahn equation
  • Gamma convergence
  • Large deviation theory
  • Sharp-interface limits
  • Stochastic partial differential equations

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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