Sharp-interface limit of the Allen-Cahn action functional in one space dimension |
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Authors: | Robert V Kohn Maria G Reznikoff Yoshihiro Tonegawa |
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Institution: | (1) Courant Institute of Mathematical Sciences, New York University, USA;(2) Institute for Applied Mathematics, University of Bonn, Germany;(3) Present address: Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA;(4) Department of Mathematics, Hokkaido University, Japan |
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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 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.
Mathematics Subject Classification (2000) 49J45, 35R60, 60F10 |
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Keywords: | Allen-Cahn equation Stochastic partial differential equations Large deviation theory Action minimization Sharp-interface limits Gamma convergence |
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