Paper · Mar 12, 2020

Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

Giovanni Bellettini, Antonin Chambolle, Shokhrukh Yu. Kholmatov

math.APWider flowsarXiv:2003.05761
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Abstract

Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global -Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean curvature two-phase flow.

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Submitted
Mar 12, 2020
Subjects
math.AP

Comments: 28 pages

Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flows

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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 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 Minimizing movements for mean curvature flow of droplets with prescribed contact angle (Giovanni Bellettini and Shokhrukh Yusufovich Kholmatov, 2016)6Bellettini 2016 Minimizing movements for mean curvature flow of partitions (Giovanni Bellettini and Shokhrukh Yusufovich Kholmatov, 2017)1Bellettini 2017 Existence and uniqueness for anisotropic and crystalline mean curvature flows (Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al., 2017)5Chambolle 2017 Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2022)7Chambolle 2022 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)3Bellettini 2023 Minimizing movements for forced anisotropic curvature flow of droplets (Shokhrukh Yu. Kholmatov, 2024)2Kholmatov 2024 Minimizing movements for the generalized power mean curvature flow (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2024)4Bellettini 2024 Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions (Giovanni Bellettini, Shokhrukh Yu. Kholmatov and Matteo Novaga, 2025)8Bellettini 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Minimizing movements for mean curvature flow of partitions

    Giovanni Bellettini and Shokhrukh Yusufovich KholmatovWider flows

    Topics Weak & singular flows · Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “partition”, “minimizing movements”, “suitable driving”

  2. Minimizing movements for forced anisotropic curvature flow of droplets

    Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Shokhrukh Yu. Kholmatov · Shared terms “forced anisotropic”, “minimizing movements”, “Hölder continuous”

  3. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows · Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “partition”, “anisotropy”

  4. Minimizing movements for the generalized power mean curvature flow

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Weak & singular flows · Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “minimizing movements”, “driving force”, “anisotropic”

  5. Existence and uniqueness for anisotropic and crystalline mean curvature flows

    Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Antonin Chambolle · Shared terms “mobility”, “minimizing movements”, “anisotropy”

  6. Minimizing movements for mean curvature flow of droplets with prescribed contact angle

    Giovanni Bellettini and Shokhrukh Yusufovich KholmatovWider flows

    Topics Weak & singular flows · Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “minimizing movements”

  7. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows

    Topics Weak & singular flows · Same author Antonin Chambolle · Shared terms “minimizing movements”, “mobility”, “anisotropic”

  8. Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions

    Giovanni Bellettini, Shokhrukh Yu. Kholmatov and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Same authors Giovanni Bellettini, Shokhrukh Yu. Kholmatov · Shared terms “anisotropy”

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) 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)

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.