Advances in Nonlinear Analysis (Aug 2022)

The evolution of immersed locally convex plane curves driven by anisotropic curvature flow

  • Wang Yaping,
  • Wang Xiaoliu

DOI
https://doi.org/10.1515/anona-2022-0245
Journal volume & issue
Vol. 12, no. 1
pp. 117 – 131

Abstract

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In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V=1αψ(x)καV=\frac{1}{\alpha }\psi \left(x){\kappa }^{\alpha } for α1\alpha \gt 1, where x∈[0,2mπ]x\in \left[0,2m\pi ] is the tangential angle at the point on evolving curves. For −1≤α1\alpha \gt 1, we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function ψ\psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.

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