Paper · Feb 4, 2017

Minimizing movements for mean curvature flow of partitions

Giovanni Bellettini, Shokhrukh Yusufovich Kholmatov

math.APWider flowsarXiv:1702.01322v2
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Abstract

We prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some consistency results for a minimizing movement solution with smooth and viscosity solutions when the evolution starts from a partition made by a union of bounded sets at a positive distance. In addition, the motion starting from the union of convex sets at a positive distance agrees with the classical mean curvature flow and is stable with respect to the Hausdorff convergence of the initial partitions.

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Submitted
Feb 4, 2017
Last revised
May 16, 2018 (v2)
Subjects
math.AP

Comments: 26 pages

Mean curvature flowWeak & singular flowsLong-time existence & convergence

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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 Minimizing movements for mean curvature flow of droplets with prescribed contact angle (Giovanni Bellettini and Shokhrukh Yusufovich Kholmatov, 2016)3Bellettini 2016 Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities (Giovanni Bellettini, Antonin Chambolle and Shokhrukh Yu. Kholmatov, 2020)1Bellettini 2020 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)5Bellettini 2023 Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in ℝ³ (Shokhrukh Kholmatov, 2024)4Kholmatov 2024 Minimizing movements for forced anisotropic curvature flow of droplets (Shokhrukh Yu. Kholmatov, 2024)6Kholmatov 2024 Minimizing movements for the generalized power mean curvature flow (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2024)2Bellettini 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 forced anisotropic mean curvature flow of partitions with mobilities

    Giovanni Bellettini, Antonin Chambolle and Shokhrukh Yu. KholmatovWider flows

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

  2. 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 Yusufovich Kholmatov · Shared terms “minimizing movements”, “driving force”

  3. 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 Yusufovich Kholmatov · Shared terms “minimizing movements”

  4. Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in

    Shokhrukh KholmatovWider flows

    Topics Weak & singular flows · Same author Shokhrukh Yusufovich Kholmatov · Shared terms “minimizing movements”, “movement solution”, “consistency”

  5. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Same authors Giovanni Bellettini, Shokhrukh Yusufovich Kholmatov · Shared terms “partition”

  6. Minimizing movements for forced anisotropic curvature flow of droplets

    Shokhrukh Yu. KholmatovWider flows

    Topics Weak & singular flows · Same author Shokhrukh Yusufovich Kholmatov · Shared terms “minimizing movements”, “movement solution”, “consistency”

  7. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows

    Topics Weak & singular flows · Shared terms “minimizing movements”, “viscosity solutions”

  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 Long-time existence & convergence · Same authors Giovanni Bellettini, Shokhrukh Yusufovich Kholmatov

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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.