Paper · Jan 2, 2022

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

Jakob Fuchs, Tim Laux

math.APWider flowsarXiv:2201.00413v3
PDF Abstract page

As on arXiv

Abstract

In this work, we analyze Merriman, Bence and Osher's thresholding scheme, a time discretization for mean curvature flow. We restrict to the two-phase setting and mean convex initial conditions. In the sense of the minimizing movements interpretation of Esedoglu and Otto we show the time-integrated energy of the approximation to converge to the time-integrated energy of the limit. As a corollary, the conditional convergence results of Otto and one of the authors become unconditional in the two-phase mean convex case. Our results are general enough to handle the extension of the scheme to anisotropic flows for which a non-negative kernel can be chosen.

Comments: 59 pages, 12 figures, revised version, additional examples and discussion

Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flowsPhase fields & multiphase flows

Its neighbourhood

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 2021, 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 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)5Hensel 2021 BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)8Hensel 2021 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)2Laux 2022 A mean curvature flow arising in adversarial training (Leon Bungert, Tim Laux and Kerrek Stinson, 2024)4Bungert 2024 Median filter method for mean curvature flow using a random Jacobi algorithm (Anton Ullrich and Tim Laux, 2024)1Ullrich 2024 Convergence of thresholding energies for anisotropic mean curvature flow on inhomogeneous obstacle (Andrea Chiesa and Karel Svadlenka, 2025)3Chiesa 2025 Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds (Fabius Krämer, 2025)6Krämer 2025 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. Median filter method for mean curvature flow using a random Jacobi algorithm

    Anton Ullrich and Tim LauxWider flows PDF

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

  2. Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow

    Tim Laux, Kerrek Stinson and Clemens UllrichWider flows PDF

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

  3. Convergence of thresholding energies for anisotropic mean curvature flow on inhomogeneous obstacle

    Andrea Chiesa and Karel SvadlenkaWider flows PDF

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows · Shared terms “thresholding”, “Otto”, “Esedoglu”

  4. A mean curvature flow arising in adversarial training

    Leon Bungert, Tim Laux and Kerrek StinsonWider flows PDF

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

  5. 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 PDF

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

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

    Fabius KrämerWider flows PDF

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “thresholding scheme”, “Otto”, “time discretization”

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

    Tim Laux and Keisuke TakasaoWider flows PDF

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

  8. BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

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

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

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

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.