Mugnai, Luca; Röger, Matthias The Allen-Cahn action functional in higher dimensions. (English) Zbl 1288.93096 Interfaces Free Bound. 10, No. 1, 45-78 (2008). Summary: 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. Cited in 19 Documents MSC: 93E20 Optimal stochastic control 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 49J45 Methods involving semicontinuity and convergence; relaxation 35R60 PDEs with randomness, stochastic partial differential equations 35K55 Nonlinear parabolic equations Keywords:Allen-Cahn equation; stochastic partial differential equations; large deviation theory; sharp interface limits; motion by mean curvature PDFBibTeX XMLCite \textit{L. Mugnai} and \textit{M. Röger}, Interfaces Free Bound. 10, No. 1, 45--78 (2008; Zbl 1288.93096) Full Text: DOI arXiv