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This paper studies a fourth-order crystalline curvature ow for a curve represented by the graph of a spatially periodic function. This is a spe- cial 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 firm. Moreover, an example of facet-merging is given as well as several estimates for the speed of each facet.

This paper solves a singular initial value problem for a system of ordinary di erential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.

A planar anisotropic curvature flow equation with constant driving force term is considered when the interfacial energy is crystalline. The driving force term is given so that a closed convex set grows if it is sufficiently large. If initial shape is convex, it is shown that a flat part called a facet (with admissible orientation) is instantaneously formed. Moreover, if the initial shape is convex and slightly bigger than the critical size, the shape becomes fully faceted in a finite time provided that the Frank diagram of interfacial energy density is a regular polygon centered at the origin. The proofs of these statements are based on approximation by crystalline algorithm whose foundation was established a decade ago. Our results indicate that the anisotropy of intefacial energy plays a key role when crystal is small in the theory of crystal growth. In particular, our theorems explain a reason why snow crystal forms a hexagonal prism when it is very small.

Consider the solution u(x; t) of the heat equation with initial data u0. The diffusive sign SD[u0](x) is de ned by the limit of sign of u(x; t) as t ! 0. A sufficient condition for x 2 Rd and u0 such that SD[u0](x) is well-de ned is given. A few examples of u0 violating and ful lling this condition are given. It turns out that this diffusive sign is also related to variational problem whose energy is the Dirichlet energy with a delty term. If initial data is a difference of characteristic function of two disjoint sets, it turns out that the boundary of the set SD[u0](x) = 1 (or 􀀀1) is roughly an equi-distance hypersurface from A and B and this gives a separation of two data sets.

For a given sector a selfsimilar expanding solution to a crystalline flow is constructed. The solution is shown to be unique. Because of selfsimilarity the problem is reduced to solve a system of algebraic equations of degree two. The solution is constructed by a method of continuity and obtained by solving associated ordinary differential equations. The selfsimilar expanding solution is useful to construct a crystalline flow from an arbitrary polygon not necessarily admissible.