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Sharp-interface limit of the Allen-Cahn action functional in one space dimension

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Title: Sharp-interface limit of the Allen-Cahn action functional in one space dimension
Authors: Kohn, Robert V. Browse this author
Reznikoff, Maria G. Browse this author
Tonegawa, Yoshihiro Browse this author
Keywords: Allen-Cahn equation
stochastic partial diffrential equations
large deviation theory
action minimization
sharp-interface limits
gamma convergence
Issue Date: Apr-2006
Publisher: Springer
Journal Title: Calculus of variations and partial differential equations
Volume: 25
Issue: 4
Start Page: 503
End Page: 534
Publisher DOI: 10.1007/s00526-005-0370-5
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 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, constructions 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 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.
Rights: The original publication is available at
Type: article (author version)
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

Submitter: 利根川 吉廣

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