CRYSTALLINE SURFACE DIFFUSION FLOW FOR GRAPH-LIKE CURVES

MI-HO GIGA, YOSHIKAZU GIGA
This paper studies a fourth-order crystalline curvature flow for a curve represented by the graph of a spatially periodic function. This is a special example of general crystalline surface diffusion flow. We consider a special class of piecewise linear functions and calculate its speed. We introduce notion of firmness and prove that the solution stays firm if initially it is firm at least for a short time. We also give an example that a facet (flat part) may split if the initial profile is not
more » ... irm. Moreover, an example of facet-merging is given as well as several estimates for the speed of each facet. Introduction. A surface diffusion flow equation was introduced by W. W. Mullins [23] to model motion of a phase boundary in a relaxation dynamics; see also [4] for a review. When two phases are bounded by a closed curve Γ t depending on time t in the plane R 2 , its typical form is Here, V denotes the normal velocity of Γ t in the direction of unit normal n of Γ t and κ denotes the curvature in the direction of n. The subscript s denotes the derivative with respect to arclength so that κ ss = ∆ Γt κ, where ∆ Γ denotes the Laplace-Beltrami operator on a curve Γ. (Its higher dimensional version is V = −∆ Γ κ, where κ denotes the mean curvature of an evolving hypersurface Γ t in the direction of n in R d .) This equation is considered as a gradient flow of the
doi:10.14943/100842 fatcat:sqwob4fdubecpm2ve466k33pxi