The Thresholding Scheme for Mean Curvature Flow and de Giorgi’s Ideas for Gradient Flows

Flow of interfaces by mean curvature, in its multi-phase version, was first formulated in the context of grain growth in polycrystalline materials. The computationally efficient and very popular thresholding scheme for mean curvature flow by Osher et. al. can be naturally extended to such a multi-phase situation, even for surface tensions that depend on the lattice mismatch between the adjacent grains.

This extension relies on the gradient flow structure of mean curvature flow, and the interpretation of the thresholding scheme as a corresponding "minimizing movements'' scheme, that is, a sequence of variational problems naturally attached to the implicit time discretization of a gradient flow.

This interpretation also allows for a (conditional) convergence proof based on De Giorgi's ideas for gradient flows in metric spaces. The approach is similar to the convergence proof for the minimizing movement scheme by Almgren, Taylor and Wang, as given by Luckhaus et. al.

This is joint work with S. Esedoglu and T. Laux.