Calculus of Variations and Geometric Measure Theory

A. Braides - M. S. Gelli - M. Novaga

Motion and pinning of discrete interfaces

created by braidesa on 07 May 2008
modified on 19 Jan 2010


Published Paper

Inserted: 7 may 2008
Last Updated: 19 jan 2010

Journal: Arch. Ration. Anal. Mech.
Volume: 95
Pages: 469-498
Year: 2010


We describe the motion of interfaces in a two-dimensional discrete environment by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuous analysis. We show that below a critical ratio of the time and space scalings we have no motion of interfaces (pinning), while above that ratio the discrete motion is approximately described by the crystalline motion by curvature on the continuum described by Almgren and Taylor. The critical regime is much richer, exhibiting a pinning threshold (small sets move, large sets are pinned), partial pinning (portions of interfaces may not move), pinning after an initial motion (possibly to a non-convex limit set), ``quantization'' of the interface velocity, and non-uniqueness effects.

Keywords: Discrete energies, anisotropic perimeter, motion by curvature, crystalline curvature