标题： 周期介质中Allen-Cahn和曲率流的表面张力梯度不连续性和零迁移率的发生 显示英文标题

标题： The occurrence of surface tension gradient discontinuities and zero mobility for Allen-Cahn and curvature flows in periodic media

摘要： 我们构建了与异质介质中能量极小化器和表面能泛函的梯度流的尺度极限相关的几个例子。 这些包括尖锐界面模型和扩散界面模型。 重点是两个独立但相关的问题，有效表面张力的正则性和相关梯度流中零迁移率的发生。 在正则性方面，我们基于Goldman、Chambolle和Novaga的理论，证明在尖锐界面模型中，表面张力的梯度不连续性是普遍存在的。 在扩散界面情况下，我们仅证明由满足强Birkhoff性质的平面类似解组成的层状结构通常是非叶状结构，并且存在间隙。 在迁移率方面，我们在尖锐界面和扩散界面情况下都构造了例子，其中$L^2$梯度流的均质化尺度极限是平凡的，即在每个方向上都有钉扎现象。 在尖锐界面情况下，这些例子与Novaga和Valdinoci之前为强制平均曲率流构造的例子有关。 显示英文摘要

摘要： We construct several examples related to the scaling limits of energy minimizers and gradient flows of surface energy functionals in heterogeneous media. These include both sharp and diffuse interface models. The focus is on two separate but related issues, the regularity of effective surface tensions and the occurrence of zero mobility in the associated gradient flows. On regularity we build on the theory of Goldman, Chambolle and Novaga to show that gradient discontinuities in the surface tension are generic for sharp interface models. In the diffuse interface case we only show that the laminations by plane-like solutions satisfying the strong Birkhoff property generically are not foliations and do have gaps. On mobility we construct examples in both the sharp and diffuse interface case where the homogenization scaling limit of the $L^2$ gradient flow is trivial, i.e. there is pinning at every direction. In the sharp interface case, these are related to examples previously constructed by Novaga and Valdinoci for forced mean curvature flow.