On a gradient flow of plane curves minimizing the anisoperimetric ratio

Details On a gradient flow of plane curves minimizing the anisoperimetric ratio On a gradient flow of plane curves minimizing the anisoperimetric ratio

2012-03-10

arxiv.org/abs/1203.2238v1

Mathematics

Differential Geometry

Scientific paper

We analyze a gradient flow of closed planar curves minimizing the anisoperimetric ratio. For such a flow the normal velocity is a function of the anisotropic curvature and it also depends on the total interfacial energy and enclosed area of the curve. In contrast to the gradient flow for the isoperimetric ratio, we show there exist initial curves for which the enclosed area is decreasing with respect to time. We also derive a mixed anisoperimetric inequality for the product of total interfacial energies corresponding to different anisotropy functions. Finally, we present several computational examples illustrating theoretical results.

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