# American Institute of Mathematical Sciences

November  2016, 36(11): 6379-6411. doi: 10.3934/dcds.2016076

## Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature

 1 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, United States 2 Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907

Received  January 2015 Revised  July 2016 Published  August 2016

We analyze the continuum limit of a thresholding algorithm for motion by mean curvature of one dimensional interfaces in various space-time discrete regimes. The algorithm can be viewed as a time-splitting scheme for the Allen-Cahn equation which is a typical model for the motion of materials phase boundaries. Our results extend the existing statements which are applicable mostly in semi-discrete (continuous in space and discrete in time) settings. The motivations of this work are twofolds: to investigate the interaction between multiple small parameters in nonlinear singularly perturbed problems, and to understand the anisotropy in curvature for interfaces in spatially discrete environments. In the current work, the small parameters are the spatial and temporal discretization step sizes: $\triangle x = h$ and $\triangle t = \tau$. We have identified the limiting description of the interfacial velocity in the (i) sub-critical ($h \ll \tau$), (ii) critical ($h = O(\tau)$), and (iii) super-critical ($h \gg \tau$) regimes. The first case gives the classical isotropic motion by mean curvature, while the second produces intricate pinning and de-pinning phenomena, and anisotropy in the velocity function of the interface. The last case produces no motion (complete pinning).
Citation: Oleksandr Misiats, Nung Kwan Yip. Convergence of space-time discrete threshold dynamics to anisotropic motion by mean curvature. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6379-6411. doi: 10.3934/dcds.2016076
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