Calculus of Variations and Geometric Measure Theory
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A. Cesaroni - L. De Luca - M. Novaga - M. Ponsiglione

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 28 Dec 2020


Accepted Paper

Inserted: 4 mar 2020
Last Updated: 28 dec 2020

Journal: Comm. Partial Differential Equations
Year: 2020

ArXiv: 2003.02248 PDF


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.


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