Paper · Oct 24, 2019

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

Tim Laux, Felix Otto

math.APWider flowsarXiv:1910.11442
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Abstract

We consider the thresholding scheme and explore its connection to De Giorgi's ideas on gradient flows in metric spaces; here applied to mean curvature flow as the steepest descent of the interfacial area. The basis of our analysis is the observation by Esedoglu and the second author that thresholding can be interpreted as a minimizing movements scheme for an energy that approximates the interfacial area. De Giorgi's framework provides an optimal energy dissipation relation for the scheme in which we pass to the limit to derive a dissipation-based weak formulation of mean curvature flow. Although applicable in the general setting of arbitrary networks, here we restrict ourselves to the case of a single interface, which allows for a compact, self-contained presentation.

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Submitted
Oct 24, 2019
Subjects
math.AP

Comments: 27 pages

Mean curvature flowWeak & singular flowsPhase fields & multiphase flows

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Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Convergence of the thresholding scheme for multi-phase mean-curvature flow (Tim Laux and Felix Otto, 2016)1Laux 2016 The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions (Julian Fischer, Sebastian Hensel, Tim Laux et al., 2020)4Fischer 2020 A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)8Hensel 2021 Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets (Jakob Fuchs and Tim Laux, 2022)2Fuchs 2022 Median filter method for mean curvature flow using a random Jacobi algorithm (Anton Ullrich and Tim Laux, 2024)3Ullrich 2024 Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds (Fabius Krämer, 2025)5Krämer 2025 De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach (Tim Laux and Andrea Poiatti, 2026)6Laux 2026 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)7Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of the thresholding scheme for multi-phase mean-curvature flow

    Tim Laux and Felix OttoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same authors Tim Laux, Felix Otto · Shared terms “thresholding scheme”, “minimizing movements”, “interfacial”

  2. Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

    Jakob Fuchs and Tim LauxWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Tim Laux · Shared terms “thresholding scheme”, “minimizing movements”, “Esedoglu”

  3. Median filter method for mean curvature flow using a random Jacobi algorithm

    Anton Ullrich and Tim LauxWider flows

    Topics Phase fields & multiphase flows · Same author Tim Laux · Shared terms “thresholding scheme”

  4. The local structure of the energy landscape in multiphase mean curvature flow: Weak-strong uniqueness and stability of evolutions

    Julian Fischer, Sebastian Hensel, Tim Laux et al.Wider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Tim Laux · Shared terms “steepest descent”, “gradient flow”

  5. Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds

    Fabius KrämerWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “thresholding scheme”, “Giorgi ideas”, “minimizing movements”

  6. De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

    Tim Laux and Andrea PoiattiWider flows

    Topics Weak & singular flows · Same author Tim Laux · Shared terms “Giorgi”, “minimizing movements”, “movements scheme”

  7. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Tim Laux · Shared terms “interface”

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

    Sebastian Hensel and Tim LauxWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Tim Laux · Shared terms “formulation”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)

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  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

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