论文标题
具有周期性迁移率的艾伦-CAHN方程的均质化
Homogenization of the Allen-Cahn equation with periodic mobility
论文作者
论文摘要
我们通过各向异性，空间周期性的迁移率系数分析了Allen-CAHN方程的尖锐界面极限，并证明接口的大规模行为是通过具有有效迁移率的平均曲率流量来确定的。正式地，结果是由Barles和Souganidis开发的渐近反应扩散方程与周期性系数开发的。但是，我们表明，当正常方向是有理的时，相应的细胞问题实际上是错误的。为了避免此问题，在控制界面的大规模行为的介观子和超扫描中，都需要许多新的想法，以及在极限中获得的接口实际上是由有效方程式描述的。
We analyze the sharp interface limit for the Allen-Cahn equation with an anisotropic, spatially periodic mobility coefficient and prove that the large-scale behavior of interfaces is determined by mean curvature flow with an effective mobility. Formally, the result follows from the asymptotics developed by Barles and Souganidis for bistable reaction-diffusion equations with periodic coefficients. However, we show that the corresponding cell problem is actually ill-posed when the normal direction is rational. To circumvent this issue, a number of new ideas are needed, both in the construction of mesoscopic sub- and supersolutions controlling the large-scale behavior of interfaces and in the proof that the interfaces obtained in the limit are actually described by the effective equation.
