The thresholding scheme for mean curvature flow and de Giorgi's ideas for minimizing movements

De Giorgi’s inequality for the thresholding scheme with arbitrary mobilities and surface tensions

“…The convergence assumption (38) is motivated by a similar assumption on the implicit time discretization in the seminal paper [25] by Luckhaus and Sturzenhecker, and has also appeared in previous work in the context of the thresholding scheme [19][20][21]. As of now, this assumption can be verified only in particular cases, such as before the first singularity [36] or for certain types of singularities, namely mean convex ones, meaning H > 0.…”

“…Inspired by the general framework [2] and [34], generalizing the previous two-phase version [21], we propose the following definition for weak solutions in the case of multiphase mean curvature flow.…”

“…is a straight-forward generalization of the argument in [21], the main novelty of the present work lies in the sharp lower bound for the distance-term of the form…”

“…Proof For u, v ∈ M definition (28) and the fact that A and B are positive definite imply that d h is a distance on M. The fact that (M, d h ) is compact and E h is continuous is just a consequence of the fact that d h metrizes the weak convergence in L 2 on M, the interested reader may find the details of the reasoning in [21]. We are thus left with showing that χ n satisfies (29…”

“…[2]. This research direction was initiated by Otto and the first author and appeared in the lecture notes [21]. There, De Giorgi's inequality is derived for the simple model case of two phases.…”