Facet{breaking for Three{dimensional Crystals Evolving by Mean Curvature
نویسندگان
چکیده
We show two examples of facet{breaking for three{dimensional poly-hedral surfaces evolving by crystalline mean curvature. The analysis shows that creation of new facets during the evolution is a common phenomenon. The rst 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 ows 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 Wull shape.
منابع مشابه
Crystalline mean curvature flow of convex sets
We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in R . This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, sa...
متن کاملUltra-Sharpening of Diamond Stylus by 500 eV O+/O2 + Ion Beam Machining without Facet and Ripple Formation
The price of single point diamond tools with a sharp tip is very high due to complex machining process and highly expensive machining equipments. Yet, the performance is not quite satisfactory. In this paper, we have presented a very simple and cost effective machining process for the sharpening and polishing of diamond stylus using low energy reactive ion beam machining (RIBM). In our method, ...
متن کاملRelation between growth and melt shapes of ice crystals
Under near-equilibrium growth or melt conditions, ice crystals in the melt are bounded by two parallel facets (basal planes or c-facets) and rounded surfaces between the facets. We observed such disk-shaped ice crystals perpendicularly to the c-axis in two and three dimensions, and found that the growth andmelt shapes are asymmetric. On imposition of a small driving force, a rounded crystal gre...
متن کاملFlow under Curvature: Singularity Formation, Minimal Surfaces, and Geodesics
We study hypersurfaces moving under flow that depends on the mean curvature. The approach is based on a numerical technique that embeds the evolving hypersurface as the zero level set of a family of evolving surfaces. In this setting, the resulting partial differential equation for the motion of the level set function φ may be solved by using numerical techniques borrowed from hyperbolic conser...
متن کاملElastic instability of a crystal growing on a curved surface.
Although the effects of kinetics on crystal growth are well understood, the role of substrate curvature is not yet established. We studied rigid, two-dimensional colloidal crystals growing on spherical droplets to understand how the elastic stress induced by Gaussian curvature affects the growth pathway. In contrast to crystals grown on flat surfaces or compliant crystals on droplets, these cry...
متن کامل