矩形的离散与连续保面积平均曲率流
Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles
AI总结：
本研究结合约束最小化运动格式与离散到连续分析，研究二维晶格晶体的保面积平均曲率型运动，刻画了连续与离散模型的控制方程，并证明时空尺度同速率趋于零时离散到连续演化由不可退化为ODE的微分包含系统描述。
摘要：
我们研究二维晶格晶体的保面积平均曲率型运动，该模型通过将Almgren、Taylor与Wang提出的约束最小化运动格式与离散到连续分析相结合得到。我们首先考察该模型的连续对应形式，证明了源自矩形的平坦流的存在性与唯一性，并刻画了其控制常微分方程组。随后在原子尺度设定下，我们明确了时间离散流的几何性质，描述了其控制有限差分包含系统。最后，在空间与时间尺度以相同速率趋于零的极限情形下，我们证明离散到连续的演化由一个微分包含系统描述，且该系统永远不会退化为常微分方程组。
英文摘要：
We investigate the area-preserving mean-curvature-type motion of a two-dimensional lattice crystal obtained by coupling constrained minimizing movements scheme introduced by Almgren, Taylor and Wang with a discrete-to-continuous analysis. We first examine the continuum counterpart of the model and establish the existence and uniqueness of the flat flow, originating from a rectangle. Additionally, we characterize the governing system of ordinary differential equations. Subsequently, in the atomistic setting, we identify geometric properties of the discrete-in-time flow and describe the governing system of finite-difference inclusions. Finally, in the limit where both spatial and time scales vanish at the same rate, we prove that a discrete-to-continuum evolution is expressed through a system of differential inclusions which does never reduce to a system of ODEs.
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