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.
Classification :
35-XX, 65-XX, 76-XX, 92-XX
Mots-clés :
Allen–Cahn equation, stochastic partial differential equations, large deviation theory, sharp interface limits, motion by mean curvature
Affiliations des auteurs :
Luca Mugnai
1
;
Matthias Röger
2
1
Mathematik in den Naturwissenschaften, Leipzig, Germany
2
Technische Universität Dortmund, Germany
Luca Mugnai; Matthias Röger. The Allen–Cahn action functional in higher dimensions. Interfaces and free boundaries, Tome 10 (2008) no. 1, pp. 45-78. doi: 10.4171/ifb/179
@article{10_4171_ifb_179,
author = {Luca Mugnai and Matthias R\"oger},
title = {The {Allen{\textendash}Cahn} action functional in higher dimensions},
journal = {Interfaces and free boundaries},
pages = {45--78},
year = {2008},
volume = {10},
number = {1},
doi = {10.4171/ifb/179},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/179/}
}
TY - JOUR
AU - Luca Mugnai
AU - Matthias Röger
TI - The Allen–Cahn action functional in higher dimensions
JO - Interfaces and free boundaries
PY - 2008
SP - 45
EP - 78
VL - 10
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/179/
DO - 10.4171/ifb/179
ID - 10_4171_ifb_179
ER -
