Calculus of Variations and Geometric Measure Theory

A. Chambolle - M. Morini - M. Ponsiglione

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

created by ponsiglio on 15 Jan 2012
modified on 19 Jan 2012



Inserted: 15 jan 2012
Last Updated: 19 jan 2012

Year: 2012


We address in this paper the study of a geometric evolution, corresponding to a curvature which is non-local and singular at the origin. The curvature represents the first variation of the energy proposed in a recent work 5 as a variant of the standard perimeter penalization for the denoising of nonsmooth curves.

To deal with such 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, and we illustrate some examples, comparing the results with the standard mean curvature flow.

Keywords: Viscosity solutions, minimizing movements, nonlocal curvature flows, nonlocal geometric flows