脉动驻波的变分原理和锐界面极限下的Einstein关系
A Variational Principle for Pulsating Standing Waves and an Einstein Relation in the Sharp Interface Limit
摘要：
本文研究了周期性介质中Allen-Cahn能量泛函的有效大尺度行为与相关的$L^{2}$梯度流的尖锐界面极限之间的联系。通过在圆柱体$\mathbb{R} \times \mathbb{T}^{d}$中引入一个Percival型拉格朗日量，我们建立了Anisini，Braides和Chiadhesi Piat的$Γ$ -收敛结果与Barles和Souganesi的尖锐界面极限结果之间的联系。在层流介质中，我们证明了一个尖锐的界面限制在图形设置，使没有假设以外的系数足够的平滑度，我们证明了有效的界面速度和表面张力满足爱因斯坦关系。一些病理突出的困难，不出现在空间均匀的设置。
英文摘要：
This paper investigates the connection between the effective, large scale behavior of Allen-Cahn energy functionals in periodic media and the sharp interface limit of the associated $L^{2}$ gradient flows. By introducing a Percival-type Lagrangian in the cylinder $\mathbb{R} \times \mathbb{T}^{d}$, we establish a link between the $Γ$-convergence results of Anisini, Braides, and Chiadò Piat and the sharp interface limit results of Barles and Souganidis. In laminar media, we prove a sharp interface limit in a graphical setting, making no assumptions other than sufficient smoothness of the coefficients, and we prove that the effective interface velocity and surface tension satisfy an Einstein relation. A number of pathologies are presented to highlight difficulties that do not arise in the spatially homogeneous setting.
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