Paper · Mar 14, 2024

Minimizing movements for forced anisotropic curvature flow of droplets

Shokhrukh Yu. Kholmatov

math.APWider flowsarXiv:2403.09902
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Abstract

We study forced anisotropic curvature flow of droplets on an inhomogeneous horizontal hyperplane. As in [Bellettini, Kholmatov: J. Math. Pures Appl. (2018)] we establish the existence of smooth flow, starting from a regular droplet and satisfying the prescribed anisotropic Young's law, and also the existence of a -Hölder continuous in time minimizing movement solution starting from a set of finite perimeter. Furthermore, we investigate various properties of minimizing movements, including comparison principles, uniform boundedness and the consistency with the smooth flow.

Comments: 35 pages, 2 figures

Other curvature flowsAnisotropic & nonlocal flowsWeak & singular flowsHomogeneous spaces

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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 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 Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2022)3Chambolle 2022 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)2Bellettini 2023 Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in ℝ³ (Shokhrukh Kholmatov, 2024)1Kholmatov 2024 Crystalline hexagonal curvature flow of networks: short-time, long-time and self-similar evolutions (Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon Almuratov, 2024)8Bellettini 2024 Minimizing movements for the generalized power mean curvature flow (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2024)4Bellettini 2024 A mean curvature flow arising in adversarial training (Leon Bungert, Tim Laux and Kerrek Stinson, 2024)7Bungert 2024 Variational Nonlinear and Nonlocal Curvature Flows (Daniele De Gennaro, 2025)6De Gennaro 2025 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)5Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in

    Shokhrukh KholmatovWider flows PDF

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

  2. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same author Shokhrukh Yu. Kholmatov

  3. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows PDF

    Topics Homogeneous spaces, Weak & singular flows · Shared terms “minimizing movements”, “inhomogeneous”

  4. Minimizing movements for the generalized power mean curvature flow

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows PDF

    Topics Weak & singular flows · Same author Shokhrukh Yu. Kholmatov · Shared terms “minimizing movements”

  5. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows

  6. Variational Nonlinear and Nonlocal Curvature Flows

    Daniele De GennaroWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Other curvature flows · Shared terms “minimizing movements”

  7. A mean curvature flow arising in adversarial training

    Leon Bungert, Tim Laux and Kerrek StinsonWider flows PDF

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “minimizing movements”, “perimeter”

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

    Giovanni Bellettini, Shokhrukh Kholmatov and Firdavsjon AlmuratovWider flows PDF

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same author Shokhrukh Yu. Kholmatov

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Background reading

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

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) 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)

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

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