Approximation of the Gauss curvature flow by a three-dimensional crystalline motion.

*(English)*Zbl 1145.53051
Beneš, Michal (ed.) et al., Proceedings of Czech-Japanese Seminar in Applied Mathematics 2005, Kuju, Japan, September 15–18, 2005. Fukuoka: Kyushu University, The 21st Century COE Program “DMHF”. COE Lecture Note 3, 139-145 (2006).

Summary: We consider an approximation of the Gauss curvature flow in \(\mathbb R^3\) by so-called crystalline motion. Here, the Gauss curvature flow makes a smooth strictly convex surface shrink with the outward normal velocity equal to the Gauss curvature with negative sign. Crystalline motion was introduced by J. E. Taylor [in: Differential geometry. A symposium in honour of Manfredo do Carmo, Proc. Int. Conf., Rio de Janeiro/Bras. 1988, Pitman Monogr. Surv. Pure Appl. Math. 52, 321–336 (1991; Zbl 0725.53011)] and S. B. Angenent and M. E. Gurtin [Arch. Ration. Mech. Anal. 108, No. 4, 323–391 (1989; Zbl 0723.73017)] to analyze crystal growth mathematically. The most typical crystalline motion of a polygon in \(\mathbb R^2\) makes each edge of a polygon keep the same direction but move with the normal speed inversely proportional 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 the 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., K. Ishii and H. M. Soner, SIAM J. Math. Anal. 30, No. 1, 19–37 (1999; Zbl 0963.35082)].

We extend the most typical two-dimensional crystalline motion to a three-dimensional one whose Wulff shape is a convex polyhedron \((\widetilde W{}^k)\). Here the Wulff shape represents the anisotropy of the problem. This motion makes each side of a polyhedron move with the normal speed inversely proportional to its area. We prove that this crystalline motion converges to the Gauss curvature flow in \(\mathbb R^3\) under the assumptions that the polyhedron \(\widetilde W{}^k\) converges to the unit ball \(B^3\) in the Hausdorff distance and is symmetric with respect to the origin.

Ishii and Soner [loc. cit.] showed the convergence of the two-dimensional crystalline motion to the curve shortening flow by a perturbed test function method. We employ their method and facts from the theory of the Minkowski problem [e.g., R. Schneider, Convex bodies: the Brunn-Minkowski theory. Cambridge: Cambridge Univ. Press (1993; Zbl 0798.52001)] to prove our result.

For the entire collection see [Zbl 1141.65001].

We extend the most typical two-dimensional crystalline motion to a three-dimensional one whose Wulff shape is a convex polyhedron \((\widetilde W{}^k)\). Here the Wulff shape represents the anisotropy of the problem. This motion makes each side of a polyhedron move with the normal speed inversely proportional to its area. We prove that this crystalline motion converges to the Gauss curvature flow in \(\mathbb R^3\) under the assumptions that the polyhedron \(\widetilde W{}^k\) converges to the unit ball \(B^3\) in the Hausdorff distance and is symmetric with respect to the origin.

Ishii and Soner [loc. cit.] showed the convergence of the two-dimensional crystalline motion to the curve shortening flow by a perturbed test function method. We employ their method and facts from the theory of the Minkowski problem [e.g., R. Schneider, Convex bodies: the Brunn-Minkowski theory. Cambridge: Cambridge Univ. Press (1993; Zbl 0798.52001)] to prove our result.

For the entire collection see [Zbl 1141.65001].