Point-extinction and geometric expansion of solutions to a crystalline motion
Point-extinction and geometric expansion of solutions to a crystalline motion
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晶体运动解的点消光和几何展开
DOI：
10.14492/hokmj/1350911957
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发表时间：
2001
影响因子：
0.5
通讯作者：
S. Yazaki
中科院分区：
文献类型：
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作者：
S. Yazaki
。我们考虑广义晶体运动解的渐近行为，该运动描述了由非光滑界面能驱动的平面曲线的演化。我们的主要结果表明，解多边形曲线会扩展到无穷大或收缩到单个点，具体取决于初始数据的大小和驱动力项的符号。在扩展情况下，我们表明，对于驱动力项，任何重新缩放的解多边形都会收敛到 Wulff 形状的边界，因此，如果驱动力项是常数，则即使运动是各向异性的，任何解多边形都会接近扩展正多边形。我们还给出了收缩情况下消光时间的下限和上限。在附录中，我们将从数值的角度解释离散曲率和晶体曲率的概念。
. We consider the asymptotic behavior of solutions to a generalized crystalline motion which describes evolution of plane curves driven by nonsmooth interfacial energy. Our main results say that solution polygonal curves expand to infinity or shrink to a single point depending on the size of initial data and the sign of the driving force term. In the expanding case, we show that any rescaled solution polygon converges to the boundary of the Wulff shape for the driving force term and hence if the driving force term is a constant, then any solution polygon approaches to an expanding regular polygon even if the motion is anisotropic. We also give lower and upper bounds of the extinction time for the shrinking case. In the appendix, we shall explain the notion of a discrete curvature and crystalline curvature from a numerical point of view.