On Area-preserving and Length-preserving Nonlocal Flow of Convex Closed Plane Curves

摘要

For any $\alpha>0,$ we study $k^{\alpha}$-type length-preserving andarea-preserving nonlocal flow of convex closed plane curves and show that thesetwo types of flow evolve such curves into round circles in $C^{\infty}%$-norm.$\ $Other relevant $k^{\alpha}$-type nonlocal flow is also discussedwhen $\alpha\geq1.\ $