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

Marco Cicalese, Andrea Kubin

Abstract

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.

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

Abstract

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.
Paper Structure (8 sections, 13 theorems, 352 equations, 11 figures)

This paper contains 8 sections, 13 theorems, 352 equations, 11 figures.

Table of Contents

  1. Notation
  2. The continuum setting
  3. Approximate flat $\mathrm{P}_{\vert \cdot \vert_1}$ area-preserving mean-curvature flow
  4. Flat $\mathrm{P}_{\vert \cdot \vert_1}$ area-preserving mean-curvature flow
  5. The discrete setting
  6. Discrete Steiner-like rearrangement of a set with respect to a quasi rectangle
  7. The incremental problem
  8. Flat $P_{|\cdot|_1}$ area-preserving mean-curvature flow of a quasi rectangle in $\varepsilon \mathbb{Z}^2$

Key Result

Proposition 2.3

Let $E \subset \mathbb{R}^2$ be a measurable set. The following statements hold true.

Figures (11)

  • Figure 1: The rectangle $R$, $\tilde{R}$, $R_1$, and $R_2$.
  • Figure 2: Nel lato sinistro il caso $x \leq \frac{b}{2}$ e nel lato destro il caso $x > \frac{b}{2}$.
  • Figure 3: The quasi-rectangle $QR= R \cup Q \in \mathcal{QR}_\varepsilon$
  • Figure 4: The quasi-rectangle $QR$ and the set $E$
  • Figure 5: The quasi-rectangle $QR$ and the set $E'$
  • ...and 6 more figures

Theorems & Definitions (40)

  • Definition 2.1: Approximate flat $\mathrm{P}_{\vert \cdot \vert_1}$ area-preserving mean-curvature flow
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • ...and 30 more