Crystalline surface diffusion flow for graph-like curves

Abstract

This paper studies a fourth-order crystalline curvature flow equation for a graph-like curve. The equation is a special example of general crystalline surface diffusion flow. We consider a special class of piecewise linear periodic functions and calculate its speed. We introduce the notion of firmness and prove that the solution stays firm at least for a short time for firm initial data. We show that a facet (flat part) may split if the initial profile is not firm by giving an explicit example. Moreover, we give an example of facet-merging as well as several estimates for the speed of each facet.