Paper · Feb 25, 2024

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

Tokuhiro Eto, Yoshikazu Giga

math.APWider flowsarXiv:2402.16180
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Abstract

We consider a minimizing movement scheme of Chambolle type for the mean curvature flow equation with prescribed contact angle condition in a smooth bounded domain in (). We prove that an approximate solution constructed by the proposed scheme converges to the level-set mean curvature flow with prescribed contact angle provided that the domain is convex and that the contact angle is away from zero under some control of derivatives of given prescribed angle. We actually prove that an auxiliary function corresponding to the scheme uniformly converges to a unique viscosity solution to the level-set equation with an oblique derivative boundary condition corresponding to the prescribed boundary condition.

Mean curvature flowWeak & singular flowsBoundaries & non-compactFree 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 A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder (Zhenghuan Gao, Bendong Lou and Jinju Xu, 2022)5Gao 2022 A level-set method for a mean curvature flow with a prescribed boundary (Xingzhi Bian, Yoshikazu Giga and Hiroyoshi Mitake, 2023)4Bian 2023 Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in ℝ³ (Shokhrukh Kholmatov, 2024)2Kholmatov 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)8Jang 2025 Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions (Can Cui and Nung Kwan Yip, 2025)6Cui 2025 Two dimensional anisotropic mean curvature flow with contact angle condition (Can Cui and Nung Kwan Yip, 2025)7Cui 2025 Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain (Tokuhiro Eto and Jiwoong Jang, 2026)1Eto 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, Boundaries & non-compact · Same author Tokuhiro Eto · Shared terms “contact angle”, “prescribed contact”, “movement scheme”

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

    Shokhrukh KholmatovWider 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”, “angle condition”

  4. A level-set method for a mean curvature flow with a prescribed boundary

    Xingzhi Bian, Yoshikazu Giga and Hiroyoshi MitakeWider flows PDF

    Topics Weak & singular flows · Same author Yoshikazu Giga · Shared terms “prescribed boundary”, “level set”, “set mean”

  5. 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”, “prescribed angle”

  6. Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

    Can Cui and Nung Kwan YipWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “contact angle”, “prescribed contact”, “boundary condition”

  7. Two dimensional anisotropic mean curvature flow with contact angle condition

    Can Cui and Nung Kwan YipWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “contact angle”, “angle condition”, “boundary condition”

  8. 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”, “approximate solution”, “angle condition”

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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.