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Flat flow is motion by crystalline curvature for curves with crystalline energies. (English) Zbl 0867.58020
It is shown that, for interface energies which are crystalline, motion of curves in the plane by crystalline curvature typically coincides with their flat curvature flow. An interface energy function is called crystalline in the case that its equilibrium crystal shape (Wulff shape) is a polygon. Motion by crystalline curvature results from integration of coupled systems of ordinary differential equations as set forth independently in [S. Angenent and M. E. Gurtin, II: Arch. Ration. Mech. Anal. 108, No. 4, 323-391 (1989; Zbl 0723.73017) and J. E. Taylor, Proc. Symp. Pure Math. 54, 417-438 (1993; Zbl 0823.49028)]. Flat curvature flows are the limits of sequences of variational minimizations as set forth in [F. Almgren, J. E. Taylor, and L. Wang, Curvature driven flows: A variational approach, SIAM J. Control Optimization 31, No. 2, 387-438 (1993; Zbl 0783.35002)].

MSC:
58E30 Variational principles in infinite-dimensional spaces
49Q20 Variational problems in a geometric measure-theoretic setting
53C65 Integral geometry
37C10 Dynamics induced by flows and semiflows
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