The Allen-Cahn action functional in higher dimensions
Luca Mugnai and Matthias Röger
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Submission date: 16. Apr. 2007 (revised version: July 2007)
published in: Interfaces and free boundaries, 10 (2008) 1, p. 45-78
DOI number (of the published article): 10.4171/IFB/179
MSC-Numbers: 49J45, 35R60, 60F10, 53C44
Keywords and phrases: Allen-Cahn equation, large deviation theory, sharp interface limits
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The Allen-Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen-Cahn equation. Formal calculations suggest a reduced action functional in the sharp interface limit. We prove in two and three space dimensions the corresponding lower bound. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the limit of diffuse surface area measures and introduce a generalized velocity and generalized reduced action functional in a class of evolving measures. As a corollary we obtain the Gamma convergence of the action functional in a class of regularly evolving hypersurfaces.