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.

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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