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Motion of curves by crystalline curvature, including triple junctions and boundary points. (English) Zbl 0823.49028
Greene, Robert (ed.) et al., Differential geometry. Part 1: Partial differential equations on manifolds. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 1, 417-438 (1993).
We define and lay out a theory for moving polygonal curves or networks of curves by their crystalline curvature. Crystalline curvature is defined and is analogous to ordinary curvature, in that it is the rate of change of “surface” energy with “volume” under certain deformations, as the height of these deformations goes to zero. (Here, we are in 2-space, so “surfaces” are curves and “volume” is 2-dimensional area.) Motion of polyhedral curves by crystalline curvature is analogous to the motion of curves by curvature, not only in that the normal velocity is proportional to the crystalline analog of curvature, but also in that it also satisfies a certain variational formulation.
For the entire collection see [Zbl 0773.00022].

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
35G25 Initial value problems for nonlinear higher-order PDEs
74A15 Thermodynamics in solid mechanics
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