Calculus of Variations and Geometric Measure Theory

L. Mugnai - M. Röger

The Allen-Cahn action functional in higher dimensions

created by mugnail on 16 Apr 2007
modified on 30 Jul 2007


Submitted Paper

Inserted: 16 apr 2007
Last Updated: 30 jul 2007

Year: 2007


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 the corresponding lower bound in two and three space dimensions. 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.

Keywords: Gamma-convergence, stochastic PDE, large deviation theory, motion by mean curvature