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

Laux, Tim; Otto, Félix

doi:10.2969/aspm/08510063

Cited by 6 publications

(36 citation statements)

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“…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.…”

Section: Statement Of Resultsmentioning

confidence: 97%

“…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.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…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…”

Section: Introductionmentioning

confidence: 87%

“…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…”

Section: Lemma 2 the Pair (M D H ) Is A Compact Metric Space The Function E H Is Continuous With Respect To D H For Everymentioning

confidence: 99%

“…[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.…”

Section: Introductionmentioning

confidence: 99%

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

Laux

Lelmi

2022

Calc. Var.

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We provide a new convergence proof of the celebrated Merriman–Bence–Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoḡlu and Otto and De Giorgi’s general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.

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Section: Statement Of Resultsmentioning

confidence: 97%

Section: Statement Of Resultsmentioning

confidence: 99%

“…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…”

Section: Introductionmentioning

confidence: 87%

Section: Lemma 2 the Pair (M D H ) Is A Compact Metric Space The Function E H Is Continuous With Respect To D H For Everymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Laux

Lelmi

2022

Calc. Var.

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

Laux¹,

Lelmi²

2021

Preprint

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We provide a new convergence proof of the celebrated Merriman-Bence-Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoglu and Otto and De Giorgi's general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.

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A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

Hensel,

Laux

2021

Preprint

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We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478]-and hence any classical solution to mean curvature flow-is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally nonunique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.

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

Cited by 6 publications

References 0 publications

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

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

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

A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

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