A Nonlocal Mean Curvature Flow and Its Semi-implicit Time-Discrete Approximation

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 Mρ(E) defined in (1.1), 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. We illustrate this with some examples, comparing the results with the standard mean curvature flow.