$K$ mean-convex and $K$-outward minimizing sets

TL;DR

This paper studies the evolution of sets under nonlocal mean curvature flow, focusing on the preservation of mean convexity and outward minimality, and analyzes convergence properties of the flow using level set and minimizing movement methods.

Contribution

It introduces a framework for analyzing nonlocal mean curvature flow, demonstrating preservation of geometric properties and convergence of nonlocal perimeters.

Findings

Preservation of mean convexity and outward minimality during the flow

Convergence of nonlocal perimeters of discrete evolutions to the continuous limit

Application of level set and minimizing movement schemes to nonlocal flows

Abstract

We consider the evolution of sets by nonlocal mean curvature and we discuss the preservation along the flow of two geometric properties, which are the mean convexity and the outward minimality. The main tools in our analysis are the level set formulation and the minimizing movement scheme for the nonlocal flow. When the initial set is outward minimizing, we also show the convergence of the (time integrated) nonlocal perimeters of the discrete evolutions to the nonlocal perimeter of the limit flow.