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.

No abstract

Abstract. We give a rigorous proof of the approximability of the so-called Helfrich's functional via diffuse interfaces, under a constraint on the ratio between the bending rigidity and the Gauss-rigidity.

Abstract. We prove a result concerning the approximation of the elastica functional with a sequence of second order functionals, under radial symmetry assumptions. This theorem is strictly related to a conjecture of De Giorgi [8].