Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition

Nicholas C. Owen, J. Rubinstein, P. Sternberg
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引用次数: 77

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.
狄利克雷条件下奇摄动双稳定势的极小值和梯度流
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。
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