Fractional mean curvature flows

Cyril Imbert (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

This paper is concerned with the study of a geometric flow whose law involves a singular integral operator. Such an operator is used to define a non-local mean curvature of a set and for this reason, the flow is referred to as fractional. Such a flow appears in two important applications: dislocation dynamics and phasefield theory for fractional reaction-diffusion equations. It is first defined by using the level-set method. It is proved that it can be also defined in terms of generalized flows (Barles, Souganidis, 1998) so that phasefield theory for fractional reaction-diffusion can be treated (see the working paper of the author and Souganidis).

Mots clés: fractional mean curvature flows; singular dislocation dynamics; level-set approach; stability results; comparison principles

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert. Fractional mean curvature flows. 2008. ⟨hal-00262386v2⟩