Paper · Mar 26, 2025

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

Andrea Chiesa, Karel Svadlenka

math.APWider flowsarXiv:2503.20524
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Abstract

We extend the analysis by Esedoglu and Otto (2015) of thresholding energies for the celebrated multiphase Bence-Merriman-Osher algorithm for computing mean curvature flow of interfacial networks, to the case of differing space-dependent anisotropies. In particular, we address the special setting of an obstacle problem, where anisotropic particles move on an inhomogeneous substrate. By suitable modification of the surface energies we construct an approximate energy that uses a single convolution kernel and is monotone with respect to the convolution width. This allows us to prove that the approximate energies -converge to their sharp interface counterpart.

Mean curvature flowAnisotropic & nonlocal flowsHomogeneous spacesPhase fields & multiphase flows

Its neighbourhood

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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 Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets (Jakob Fuchs and Tim Laux, 2022)1Fuchs 2022 Diffuse-interface approximation and weak-strong uniqueness of anisotropic mean curvature flow (Tim Laux, Kerrek Stinson and Clemens Ullrich, 2022)2Laux 2022 Stability analysis for the anisotropic curve shortening flow of planar networks (Michael Gößwein, Matteo Novaga and Paola Pozzi, 2023)3Gößwein 2023 Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions (Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon Almuratov, 2024)6Bellettini 2024 Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term (Bohdan Bulanyi and Berardo Ruffini, 2025)7Bulanyi 2025 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)4Chiesa 2025 Anisotropic Parabolic Obstacle Problems and the Stefan Problem: Regularity of the Evolving Free Boundary (Ezequiel Barbosa, Rosivaldo Gonçalves and Luan de Figueiredo, 2026)5Barbosa 2026 A Phase-Field Method for Curvature Flow of Networks with Triple Junction Drag (Yuchuan Yang and Selim Esedoglu, 2026)8Yang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

    Jakob Fuchs and Tim LauxWider flows PDF

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

  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 · Shared terms “interface”

  3. Stability analysis for the anisotropic curve shortening flow of planar networks

    Michael Gößwein, Matteo Novaga and Paola PozziWider flows PDF

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows

  4. Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

    Andrea Chiesa and Keisuke TakasaoWider flows PDF

    Topics Phase fields & multiphase flows · Same author Andrea Chiesa

  5. Anisotropic Parabolic Obstacle Problems and the Stefan Problem: Regularity of the Evolving Free Boundary

    Ezequiel Barbosa, Rosivaldo Gonçalves and Luan de FigueiredoWider flows PDF

    Topics Anisotropic & nonlocal flows · Shared terms “obstacle”, “interface”, “dependent”

  6. Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions

    Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon AlmuratovWider flows PDF

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows · Shared terms “anisotropy”

  7. Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

    Bohdan Bulanyi and Berardo RuffiniWider flows PDF

    Topics Anisotropic & nonlocal flows

  8. A Phase-Field Method for Curvature Flow of Networks with Triple Junction Drag

    Yuchuan Yang and Selim EsedogluWider flows PDF

    Topics Phase fields & multiphase flows, Anisotropic & nonlocal flows · Shared terms “sharp interface”, “interfacial network”

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992) 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  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.