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

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.

Keywords:

• Fractional mean curvature flow
• fractional perimeter
• level set formulation
• local and nonlocal geometric evolutions
• Minkowski content
• riesz energy
• viscosity solutions

AMS SUBJECT CLASSIFICATIONS:

• 53C44
• 35D40
• 35K93
• 35R11

Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).