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

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Title: The sharp-interface limit of the action functional for Allen-Cahn in one space dimension
Authors: Kohn,Robert V. Browse this author
Maria G,Reznikoff Browse this author
Tonegawa, Yoshihiro Browse this author
Keywords: Allen-Cahn equation
stochastic partial di®erential equations
large deviation theory
action minimization
sharp-interface limits.
Issue Date: 2005
Journal Title: Hokkaido University Preprint Series in Mathematics
Volume: 705
Start Page: 1
End Page: 38
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. Pre- viously, heuristic arguments and numerical results have suggested that the limiting action should \count" two competing costs: the cost to nucleate interfaces and the cost to propagate them. In addition, con- structions have been used to derive an upper bound for the minimal action which was proved optimal on the level of 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 func- tional itself, which is in agreement with the heuristic picture. To do so, we characterize the sharp-interface limit of the space-time energy mea- sures. The proof relies on an extension of earlier results for the related elliptic problem.
Type: bulletin (article)
URI: http://hdl.handle.net/2115/69510
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

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