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

Abstract

For any $\alpha >0,$ we study $k^{\alpha }$-type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $C^{\infty }$-norm. Other relevant $k^{\alpha }$-type nonlocal flow is also discussed when $\alpha \ge 1.$

Published in 'Calculus of Variations and Partial Differential Equations'

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