Vol. 13, No. 7, 2020

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Convex sets evolving by volume-preserving fractional mean curvature flows

Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci

Vol. 13 (2020), No. 7, 2149–2171
Abstract

We consider the volume-preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long-time asymptotics approach round spheres. The proofs are based on a priori estimates on the inner and outer radii of the solutions.

Keywords
geometric evolution equations, fractional partial differential equations, fractional perimeter, fractional mean curvature flow, asymptotic behavior of solutions
Mathematical Subject Classification 2010
Primary: 53C44, 35R11, 35B40
Milestones
Received: 21 November 2018
Revised: 19 July 2019
Accepted: 6 September 2019
Published: 10 November 2020
Authors
Eleonora Cinti
Dipartimento di Matematica
Università degli Studi di Bologna
Bologna
Italy
Carlo Sinestrari
Dipartimento di Ingegneria Civile e Ingegneria Informatica
Università di Roma “Tor Vergata”
Rome
Italy
Enrico Valdinoci
Department of Mathematics and Statistics
University of Western Australia
Crawley, WA
Australia