On the convergence of a three-dimensional crystalline motion to Gauss curvature flow(Viscosity Solution Theory of Differential Equations and its Developments)
Takeo Ushijima, Hiroki Yagisita · Institutional Repositories DataBase (IRDB) · 2006
We consider an approximation of the Gauss curvature flow in $\mathrm{R}^{3}$ by eocalled crystalline motion.Here, the Gauss curvature flow makes a smooth strictly convec surface shrink with the outward normal velocity equals to the Gauss curvature with negative sign.The crystalline motion was introduced by Rylor [15] and Angenent&Gurtin [1] to analyze crystal growth mathematically.The most typical crystalline motion of polygon in $\mathrm{R}^{2}$ makes each edge of a polygon keep the same direction but move with the norinal speed inversely proportiond to its length Although such motion is very restrictive at first glance, it is very useful not only in the mathematical theory of crystal growth but also as a numerical method for free boundary problems.In two dimensional case, there are already many researches on the relation between the crystalline motion of polygonal curves and the curvature driven motion of curves (e.g.[12]).We extend the moet typical two dimensional crystaUine motion $\sim \mathrm{t}\mathrm{o}$ a three dimensional one whose Wulff shape i8 a convex polyhedron $(W^{k})$ .Here the Wulff shape represents the anisotropy of the problem.This motion makes each side of a polyhedron move with the normal fped inversely $\mathrm{P}$ roportional to its afea.We prove this crystalline motion converges to $\mathrm{t}_{\sim}\mathrm{h}\mathrm{e}$ Gauss curvature flow in $\mathrm{R}^{3}$ under the aesumptions that the polyhedra $W^{k}$ converges to the unit ball $B^{S}$ in the Hausdorff distance and are symmetric with respect to the origin.K. Ishii and H.M. Soner[12] showed the convergence of the two di- mensional crystalline motion to the curve shortening flow by a kind of perturbed test function methods.We employ their method to prove our result under aid from the theory of Minkowski problem (e.g.[14]).