We consider an implicit time discretization for the motion of a hypersurface driven by its anisotropic mean curvature. We prove some convergence results of the scheme under very general assumptions on the forcing term, which include in particular the case of a typical path of the Brownian motion. We compare this limit with other available solutions, whenever they are defined. As a by-product of the analysis, we also provide a simple proof of the coincidence of the limit flow with the regular evolutions, defined for small times, in the case of a regular forcing term.
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Matteo Novaga, Antonin Chambolle, Implicit time discretization of the mean curvature flow with a discontinuous forcing term. Interfaces Free Bound. 10 (2008), no. 3, pp. 283–300DOI 10.4171/IFB/190