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

Abstract

We study in this paper the geometric evolution of a set $E$, with a velocity given by a “curvature” of $\partial E$ which is nonlocal and singular at the origin, in the sense that it behaves like a power of the classical curvature. This curvature is the first variation of an energy which is proportional to the volume of the set of points at a given distance to $\partial E$, and which was proposed in a recent work of Barchiesi et al. [Multiscale Model. Simul., 8 (2010), pp. 1715--1741] as a variant of the standard perimeter penalization for the denoising of nonsmooth curves. To deal with the degeneracies of our problem, we first give an abstract existence and uniqueness result for viscosity solutions of nonlocal degenerate Hamiltonians, satisfying a suitable continuity assumption with respect to Kuratowski convergence of the level sets. This abstract setting applies to an approximated variant of our flow. Then, by the method of minimizing movements, we also build a weak solution of our curvature flow. We illustrate this with some examples and compare the results with the standard mean curvature flow.