The thresholding scheme for mean curvature flow
speaker: Felix Otto (Max Planck Institute for Mathematics in the Sciences)abstract: We consider the thresholding scheme, a practically relevant time discretization for mean curvature flow (MCF) introduced by Bence-Merriman-Osher; and prove a (conditional) convergence result in the multi-phase case. The result establishes convergence towards a weak formulation in the framework of sets of finite perimeter.
The proof is based on the interpretation of the thresholding scheme as a minimizing movement scheme, which means that the thresholding scheme preserves the structure of (multi-phase) mean curvature flow as a gradient flow w.\ r.\ t.\ the total interfacial energy. More precisely, the thresholding scheme is a minimizing movement scheme for an energy functional that Gamma-converges to the total interfacial energy (joint work with Selim Esedoglu).
Our proof is similar in spirit to the convergence result by Luckhaus-Sturzenhecker of the Almgren-Taylor-Wang scheme, a more academic minimizing movement scheme for MCF. In particular, ours is a conditional convergence result, in the sense that we assume that the energy of the approximation converges to the energy of the limit. In addition, we appeal to an argument of De Giorgi to show that the limit also satisfies Brakke's inequality, a way to encode the gradient flow structure of MCF. De Giorgi's abstract set-up of metric slope and variational interpolation for minimizing movements, as formulated by Ambrosio-Gigli-Savare, is taylor-made for this limit. This is joint work with Tim Laux.
timetable:
Fri 23 Sep, 9:00 - 9:55, Palazzo dei Congressi
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