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Facet-breaking for three-dimensional crystals evolving by mean curvature. (English) Zbl 0934.49023
Summary: We show two examples of facet-breaking for three-dimensional polyhedral surfaces evolving by crystalline mean curvature. The analysis shows that creation of new facets during the evolution is a common phenomenon. The first example is completely rigorous, and the evolution after the subdivision of one facet is explicitly computed for short times. Moreover, the constructed evolution is unique among the crystalline flows with the given initial datum. The second example suggests that curved portions of the boundary may appear even starting from a polyhedral set close to the Wulff shape.

49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q10 Optimization of shapes other than minimal surfaces
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