The Allen-Cahn Action functional in higher dimensions

Details The Allen-Cahn Action functional in higher dimensions The Allen-Cahn Action functional in higher dimensions

2007-04-16

arxiv.org/abs/0704.1954v2

Mathematics

Analysis of PDEs

33 pages, 4 figures; minor changes and additions

Scientific paper

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.

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