Paper · Jan 12, 2024

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

Shokhrukh Kholmatov

math.DGWider flowsarXiv:2401.06307v2
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Abstract

In this paper we prove that in the minimizing movement solutions for mean curvature motion of droplets, obtained in [Bellettini, Kholmatov: J. Math. Pure Appl. (2018)], coincide with the smooth mean curvature flow of droplets with a prescribed (possibly nonconstant) contact angle.

Comments: 19 pages, 1 figures

Mean curvature flowWeak & singular flowsFree boundary & capillary

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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 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 A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder (Zhenghuan Gao, Bendong Lou and Jinju Xu, 2022)7Gao 2022 A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain (Tokuhiro Eto and Yoshikazu Giga, 2024)2Eto 2024 Minimizing movements for forced anisotropic curvature flow of droplets (Shokhrukh Yu. Kholmatov, 2024)1Kholmatov 2024 Minimizing movements for the generalized power mean curvature flow (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2024)5Bellettini 2024 Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization (Kiichi Tashiro, 2024)3Tashiro 2024 Existence of flat flows for volume-preserving mean curvature flow with contact angle (Jiwoong Jang, 2025)6Jang 2025 Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain (Tokuhiro Eto and Jiwoong Jang, 2026)4Eto 2026
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 curvature flow of droplets

    Shokhrukh Yu. KholmatovWider flows PDF

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

  2. A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

    Tokuhiro Eto and Yoshikazu GigaWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “prescribed contact”, “contact angle”, “minimizing movements”

  3. Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization

    Kiichi TashiroWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “prescribed contact”, “contact angle”

  4. Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain

    Tokuhiro Eto and Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “contact angle”, “minimizing movements”, “prescribed contact”

  5. Minimizing movements for the generalized power mean curvature flow

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows PDF

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

  6. Existence of flat flows for volume-preserving mean curvature flow with contact angle

    Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “contact angle”

  7. A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder

    Zhenghuan Gao, Bendong Lou and Jinju XuWider flows PDF

    Topics Free boundary & capillary · Shared terms “prescribed contact”, “contact angle”

  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 Free boundary & capillary, Weak & singular flows · Shared terms “contact angle”

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) 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

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.