# Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows

created by cesaroni on 04 Mar 2020
modified by novaga on 30 Jun 2021

[BibTeX]

Published Paper

Inserted: 4 mar 2020
Last Updated: 30 jun 2021

Journal: Comm. Partial Differential Equations
Volume: 46
Number: 7
Pages: 1344-1371
Year: 2021

ArXiv: 2003.02248 PDF

Abstract:

We introduce a notion of uniform convergence for local and nonlocal curvatures. Then, we propose an abstract method to prove the convergence of the corresponding geometric flows, within the level set formulation. We apply such a general theory to characterize the limits of $s$-fractional mean curvature flows as $s\to 0^+$ and $s\to 1^-$. In analogy with the $s$-fractional mean curvature flows, we introduce the notion of $s$-Riesz curvature flows and characterize its limit as $s\to 0^-$. Eventually, we discuss the limit behavior as $r\to 0^+$ of the flow generated by a regularization of the $r$-Minkowski content.