|
Hokkaido University Collection of Scholarly and Academic Papers >
Graduate School of Science / Faculty of Science >
Hokkaido University Preprint Series in Mathematics >
The sharp-interface limit of the action functional for Allen-Cahn in one space dimension
| 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 |
| Publisher: | Department of Mathematics, Hokkaido University |
| 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
|
Submitter: 数学紀要登録作業用
|
