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Sharp-interface limit of the Allen-Cahn action functional in one space dimension
| 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 www.springerlink.com |
| Relation: | http://www.springerlink.com |
| Type: | article (author version) |
| URI: | http://hdl.handle.net/2115/8556 |
| 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)
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Submitter: 利根川 吉廣
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