Calculus of Variations and Geometric Measure Theory

M. Cicalese - A. Kubin

Discrete and Continuum Area-Preserving Mean-Curvature Flow of Rectangles

created by cicalese on 09 Mar 2024



Inserted: 9 mar 2024
Last Updated: 9 mar 2024

Year: 2024


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 in \cite{ATW} 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.