Level set approach for fractional mean curvature flows

TL;DR

This paper develops a level set formulation for a non-local geometric flow involving fractional mean curvature, providing stability and comparison results relevant to dislocation dynamics and fractional reaction-diffusion models.

Contribution

It introduces a novel level set approach for fractional mean curvature flows and establishes stability and comparison principles for the associated geometric equations.

Findings

Proper level set formulation of the fractional mean curvature flow

Stability results for the geometric flow

Comparison principles for the associated equations

Abstract

This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. This operator is used to define a non-local mean curvature of a set. Moreover the associated flow appears in two important applications: dislocation dynamics and phase field theory for fractional reaction-diffusion equations. It is defined by using the level set method. The main results of this paper are: on one hand, the proper level set formulation of the geometric flow; on the other hand, stability and comparison results for the geometric equation associated with the flow.