Paper · Aug 3, 2021

Level-set forced mean curvature flow with the Neumann boundary condition

Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake, Hung Vinh Tran

math.APWider flowsarXiv:2108.01244v2Journal de Mathématiques Pures et Appliquées. Volume 168, December 2022, Pages 143-167
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Abstract

Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.

Comments: 26 pages, 3 figures

Mean curvature flowVolume-preserving & constrained flowsHomogeneous spacesBoundaries & 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 Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain (Jiwoong Jang, 2021)3Jang 2021 Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition (Jorge Lira and Fernanda Roing, 2022)8Lira 2022 A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting (Jiwoong Jang, 2023)4Jang 2023 A level-set method for a mean curvature flow with a prescribed boundary (Xingzhi Bian, Yoshikazu Giga and Hiroyoshi Mitake, 2023)6Bian 2023 Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals (Hiroyoshi Mitake and Hung V. Tran, 2023)1Mitake 2023 Quenching for axisymmetric hypersurfaces under forced mean curvature flows (Hiroyoshi Mitake, Yusuke Oka and Hung Vinh Tran, 2024)2Mitake 2024 A game approach to free boundary problems of anisotropic forced mean curvature flow equations (Takuya Sato, 2025)7Sato 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. Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals

    Hiroyoshi Mitake and Hung V. TranWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Homogeneous spaces · Same authors Hiroyoshi Mitake, Hung Vinh Tran · Shared terms “set forced”, “forced mean”, “large time”

  2. Quenching for axisymmetric hypersurfaces under forced mean curvature flows

    Hiroyoshi Mitake, Yusuke Oka and Hung Vinh TranWider flows PDF

    Topics Volume-preserving & constrained flows · Same authors Hiroyoshi Mitake, Hung Vinh Tran · Shared terms “forced mean”, “forcing term”

  3. Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain

    Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Boundaries & non-compact · Same author Jiwoong Jang · Shared terms “large time”, “level set”, “forced mean”

  4. A convergence rate of periodic homogenization for forced mean curvature flow of graphs in the laminar setting

    Jiwoong JangWider flows PDF

    Topics Volume-preserving & constrained flows · Same author Jiwoong Jang · Shared terms “forced mean”

  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, Boundaries & non-compact · Same author Jiwoong Jang · Shared terms “level set”, “Lipschitz”, “boundary condition”

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

    Xingzhi Bian, Yoshikazu Giga and Hiroyoshi MitakeWider flows PDF

    Topics Homogeneous spaces · Same author Hiroyoshi Mitake · Shared terms “level set”, “globally”

  7. A game approach to free boundary problems of anisotropic forced mean curvature flow equations

    Takuya SatoWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Shared terms “forced mean”, “level set”

  8. Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition

    Jorge Lira and Fernanda RoingWider flows PDF

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary 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) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) 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) 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)

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. 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.