Paper · Oct 29, 2022

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

Zhenghuan Gao, Bendong Lou, Jinju Xu

math.DGWider flowsarXiv:2210.16475v2
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Abstract

In this paper we consider a mean curvature flow in a high dimensional cylinder , where, is a constant, is a bounded domain in , and, for a hypersurface over , and denote its normal velocity and mean curvature, respectively. Assume the hypersurface contacts the cylinder boundary with prescribed angle . Under certain assumptions such as is strictly convex and is small, or is not necessarily convex but is sufficiently large, we derive some {\it uniform-in-time gradient bounds} for the solutions to initial boundary value problems. Then, we present a trichotomy result as well as its criterion for the asymptotic behavior of the solutions, that is, when (resp. , ), the solution converges as to a translating solution with positive speed (resp. stationary solution, a translating solution with negative speed).

Comments: 31 pages

Mean curvature flowSelf-shrinkers & translatorsFree 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 Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles (Bendong Lou and Lixia Yuan, 2022)1Lou 2022 Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in ℝ³ (Shokhrukh Kholmatov, 2024)4Kholmatov 2024 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)3Eto 2024 Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization (Kiichi Tashiro, 2024)2Tashiro 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)5Eto 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles

    Bendong Lou and Lixia YuanWider flows PDF

    Topics Self-shrinkers & translators · Same author Bendong Lou · Shared terms “translating solution”, “initial boundary”, “boundary value”

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

    Kiichi TashiroWider flows PDF

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

  3. 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 · Shared terms “prescribed contact”, “contact angle”, “prescribed angle”

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

    Shokhrukh KholmatovWider flows PDF

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

  5. 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 · Shared terms “contact angle”, “prescribed contact”, “bounded domain”

  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 · Shared terms “contact angle”, “prescribed contact”, “strictly convex”

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

    Can Cui and Nung Kwan YipWider flows PDF

    Topics Free boundary & capillary · Shared terms “contact angle”, “boundary value”, “strictly convex”

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

    Jiwoong JangWider flows PDF

    Topics Free boundary & capillary · 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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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. 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.