A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation

Antonin Chambolle (1) , Massimiliano Morini (2) , Marcello Ponsiglione (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Dipartimento di Matematica [Parma]
3 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]

We address in this paper the study of a geometric evolution of sets, corresponding to a curvature which is non-local and singular at the origin. The curvature represents the first variation of an energy which is the volume of the set of points at a given distance to the boundary of the set. It was proposed in a a recent work of Barchiesi, Kang, Le, Morini, Ponsiglione (SIAM MMS, 2010) as a variant of the standard perimeter penalization, for the denoising of nonsmooth curves. To deal with its degeneracies, we first give an abstract existence and uniqueness result for viscosity solutions of non-local degenerate Hamiltonians, satisfying suitable continuity assumption with respect to Kuratowsky convergence of the level sets. This abstract setting applies to an approximated flow. Then, by the method of minimizing movements, we also build an ''exact'' curvature flow. We illustrate this flow with some examples, comparing our results with the standard mean curvature flow.

Mots clés: nonlocal geometric flows; minimizing movements; perimeter regularization; singular curvatures; viscosity solutions

Domaines: Analyse numérique [math.NA]

Antonin Chambolle, Massimiliano Morini, Marcello Ponsiglione. A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation. 2012. ⟨hal-00662722⟩