Nonlocal Flow Driven by the Radius of Curvature with Fixed Curvature Integral

Abstract

This paper deals with a $$1/\kappa $$-type nonlocal flow for an initial convex closed curve $$\gamma _{0}\subset {\mathbb {R}}^{2}$$ which preserves the convexity and the integral$$\ \int _{X\left( \cdot ,t\right) }\kappa ^{\alpha +1}ds,\ \alpha \in \left( -\infty ,\infty \right) ,$$ of the evolving curve $$X\left( \cdot ,t\right) $$. For$$\ \alpha \in [1,\infty ),\ $$it is proved that this flow exists for all time $$t\in [0,\infty )$$ and $$X(\cdot ,t)$$ converges to a round circle in $$C^{\infty }$$ norm as $$t\rightarrow \infty $$. For $$\alpha \in \left( -\infty ,1\right) $$, a discussion on the possible asymptotic behavior of the flow is also given.