# The sharp-interface limit of the action functional for Allen-Cahn in one space dimension

Preprint Series # 705
Kohn, Robert V. and Maria G, Reznikoff and Tonegawa, Yoshihiro The sharp-interface limit of the action functional for Allen-Cahn in one space dimension. (2005);

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## Abstract

We analyze the sharp-interface limit of the action minimization problem for the stochastically perturbed Allen-Cahn equation in one space dimension. The action is a deterministic functional which is linked to the behavior of the stochastic process in the small noise limit. Previously, heuristic srguments and numerical result heve suggested that the limiting action should“count” two competing costs: the cost to nucleate interfaces and the cost to propagate them. In addition, constructions have been used to derive an upper bound for the minimal action which was proved optimal on the level of \it{scaling}. In this paper, we prove that for $(d=1)$, the upper bound achieved by the constructions is in fact sharp. Furthermore, we derive a lower bound for the functional itself, which is in agreement with the heuristic picture. To do so, we characterize the sharp-interface limit of the space-time energy measures. The proof relies on an extension of earlier results for the related elliptic problem.

Item Type: Preprint 10 Allen-Cahn equation, stochastic partial differential equations, large deviation theory, action minimization, sharp-interface limits 49-xx CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION 888