Convergence of the volume preserving fractional mean curvature flow for convex sets

2023/07/08 by Julin, Vesa, La Manna, Domenico Angelo
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.03912

Abstract

We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga \citeCN this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci \citeCSV2 imply the C^1+α-regularity of the flow and then provide a regularity argument which improves this into C^2+α-regularity of the flow. The regularity step from C^1+α into C^2+α does not rely on convexity and can probably be adopted to more general setting.

Related

• Stability of the ball under volume preserving fractional mean curvature flow
• Short time existence of the classical solution to the fractional Mean curvature flow
• On the evolution by fractional mean curvature
• Eventual Regularization of Fractional Mean Curvature Flow
• Mean curvature flow of mean convex hypersurfaces