Abstract
We introduce a three-dimensional crystalline motion whose Wulff shape is a convex polyhedron (Wk). We prove that this crystalline motion converges to the motion by Gauss curvature in ℝ3 under the assumptions that the polyhedra (Wk) converge to the unit ball B3 and are symmetric with respect to the origin.
K. Ishii and H. M. Soner showed the convergence of the two-dimensional 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.
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The authors are partially supported by Grant-in-Aid for Encouragement of Young Scientists (Ushijima: No. 16740061, Yagisita: No. 16740099).
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Ushijima, T.K., Yagisita, H. Convergence of a three-dimensional crystalline motion to Gauss curvature flow. Japan J. Indust. Appl. Math. 22, 443–459 (2005). https://doi.org/10.1007/BF03167494
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DOI: https://doi.org/10.1007/BF03167494