### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 503-534 |

Number of pages | 32 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 25 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2006 |

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### Keywords

- Action minimization
- Allen-Cahn equation
- Gamma convergence
- Large deviation theory
- Sharp-interface limits
- Stochastic partial differential equations

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

*Calculus of Variations and Partial Differential Equations*,

*25*(4), 503-534. https://doi.org/10.1007/s00526-005-0370-5

**Sharp-interface limit of the Allen-Cahn action functional in one space dimension.** / Kohn, Robert; Reznikoff, Maria G.; Tonegawa, Yoshihiro.

Research output: Contribution to journal › Article

*Calculus of Variations and Partial Differential Equations*, vol. 25, no. 4, pp. 503-534. https://doi.org/10.1007/s00526-005-0370-5

}

TY - JOUR

T1 - Sharp-interface limit of the Allen-Cahn action functional in one space dimension

AU - Kohn, Robert

AU - Reznikoff, Maria G.

AU - Tonegawa, Yoshihiro

PY - 2006/4

Y1 - 2006/4

N2 - 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.

AB - 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.

KW - Action minimization

KW - Allen-Cahn equation

KW - Gamma convergence

KW - Large deviation theory

KW - Sharp-interface limits

KW - Stochastic partial differential equations

UR - http://www.scopus.com/inward/record.url?scp=33644531554&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33644531554&partnerID=8YFLogxK

U2 - 10.1007/s00526-005-0370-5

DO - 10.1007/s00526-005-0370-5

M3 - Article

VL - 25

SP - 503

EP - 534

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 4

ER -