狄利克雷条件下奇摄动双稳定势的极小值和梯度流

Nicholas C. Owen, J. Rubinstein, P. Sternberg
{"title":"狄利克雷条件下奇摄动双稳定势的极小值和梯度流","authors":"Nicholas C. Owen, J. Rubinstein, P. Sternberg","doi":"10.1098/rspa.1990.0071","DOIUrl":null,"url":null,"abstract":"Minimizers and gradient flows are studied for the functional ∫Ω W(u) + ϵ2∣∇u∣2dx, Ω ⊆ Rn, ϵ > 0, where u satisfies a Dirichlet condition u = hϵ on ∂Ω. Here W is taken to be a double-well potential with minimum value zero attained at u = a and u = b. Questions of existence and structure of minimizers for small ϵ are resolved through the identification of a limiting variational problem, the so-called Γ-limit. A formal asymptotic solution is then constructed for the gradient flow ∂tuϵ = 2ϵ∆uϵ—ϵ-1W'(uϵ), uϵ(x, 0) = g(x), uϵ(x, t) = hϵ on ∂Ω, valid when ϵ is small. Using multiple timescales we show that fronts rapidly develop and then propagate with normal velocity ϵk, where k is mean curvature. At the intersection of a front with ∂Ω, the Dirichlet condition is shown to imply a contact angle condition for the front. This asymptotically correct evolution process represents gradient flow for the Γ-limit.","PeriodicalId":20605,"journal":{"name":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1990-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"77","resultStr":"{\"title\":\"Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition\",\"authors\":\"Nicholas C. Owen, J. Rubinstein, P. Sternberg\",\"doi\":\"10.1098/rspa.1990.0071\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Minimizers and gradient flows are studied for the functional ∫Ω W(u) + ϵ2∣∇u∣2dx, Ω ⊆ Rn, ϵ > 0, where u satisfies a Dirichlet condition u = hϵ on ∂Ω. Here W is taken to be a double-well potential with minimum value zero attained at u = a and u = b. Questions of existence and structure of minimizers for small ϵ are resolved through the identification of a limiting variational problem, the so-called Γ-limit. A formal asymptotic solution is then constructed for the gradient flow ∂tuϵ = 2ϵ∆uϵ—ϵ-1W'(uϵ), uϵ(x, 0) = g(x), uϵ(x, t) = hϵ on ∂Ω, valid when ϵ is small. Using multiple timescales we show that fronts rapidly develop and then propagate with normal velocity ϵk, where k is mean curvature. At the intersection of a front with ∂Ω, the Dirichlet condition is shown to imply a contact angle condition for the front. This asymptotically correct evolution process represents gradient flow for the Γ-limit.\",\"PeriodicalId\":20605,\"journal\":{\"name\":\"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1990-06-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"77\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1098/rspa.1990.0071\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1098/rspa.1990.0071","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 77

摘要

研究了泛函∫Ω W(u) + ϵ2∣∇u∣2dx, Ω Rn, λ > 0,其中u满足∂Ω上的Dirichlet条件u = hλ的最小化和梯度流。这里,W是在u = a和u = b处达到最小值为零的双阱势。通过识别一个极限变分问题,即Γ-limit,解决了小λ的最小值的存在性和结构问题。然后构造了一个正式的渐近解,用于梯度流∂tue_ (2e_∆ue_ -ϵ-1W’(ue_), ue_ (x, 0) = g(x), ue_ (x, t) = h_在∂Ω上,当ε很小时有效。使用多个时间尺度,我们显示锋面迅速发展,然后以正常速度ϵk传播,其中k是平均曲率。在锋面与∂Ω的交集处,狄利克雷条件暗示了锋面的接触角条件。这个渐近正确的演化过程表示Γ-limit的梯度流。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition
Minimizers and gradient flows are studied for the functional ∫Ω W(u) + ϵ2∣∇u∣2dx, Ω ⊆ Rn, ϵ > 0, where u satisfies a Dirichlet condition u = hϵ on ∂Ω. Here W is taken to be a double-well potential with minimum value zero attained at u = a and u = b. Questions of existence and structure of minimizers for small ϵ are resolved through the identification of a limiting variational problem, the so-called Γ-limit. A formal asymptotic solution is then constructed for the gradient flow ∂tuϵ = 2ϵ∆uϵ—ϵ-1W'(uϵ), uϵ(x, 0) = g(x), uϵ(x, t) = hϵ on ∂Ω, valid when ϵ is small. Using multiple timescales we show that fronts rapidly develop and then propagate with normal velocity ϵk, where k is mean curvature. At the intersection of a front with ∂Ω, the Dirichlet condition is shown to imply a contact angle condition for the front. This asymptotically correct evolution process represents gradient flow for the Γ-limit.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信