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 phasefield 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.
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Cyril Imbert, Level set approach for fractional mean curvature flows. Interfaces Free Bound. 11 (2009), no. 1, pp. 153–176DOI 10.4171/IFB/207