The Allen-Cahn action functional in higher dimensions. (English) Zbl 1288.93096
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.
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 |