Paper · Jun 25, 2023

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

Xingzhi Bian, Yoshikazu Giga, Hiroyoshi Mitake

math.APWider flowsarXiv:2306.14218
PDF Abstract page

As on arXiv

Abstract

We propose a level-set method for a mean curvature flow whose boundary is prescribed by interpreting the boundary as an obstacle. Since the corresponding obstacle problem is globally solvable, our method gives a global-in-time level-set mean curvature flow under a prescribed boundary with no restriction of the profile of an initial hypersurface. We show that our solution agrees with a classical mean curvature flow under the Dirichlet condition. We moreover prove that our solution agrees with a level-set flow under the Dirichlet condition constructed by P. Sternberg and W. P. Ziemer (1994), where the initial hypersurface is contained in a strictly mean-convex domain and the prescribed boundary is on the boundary of the domain.

Comments: 26 pages

Mean curvature flowWeak & singular flowsHomogeneous spacesLong-time existence & convergence

Its neighbourhood

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 Level-set forced mean curvature flow with the Neumann boundary condition (Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al., 2021)2Jang 2021 Generic regularity of Level Set Flows with spherical singularity (Ao Sun and Jinxin Xue, 2022)8Sun 2022 Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2022)4Chambolle 2022 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)6Chodosh 2023 Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals (Hiroyoshi Mitake and Hung V. Tran, 2023)3Mitake 2023 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)1Eto 2024 Generic mean curvature flow with obstacles (Tim Laux and Keisuke Takasao, 2025)5Laux 2025 Well-posedness for the mean curvature flow on the half-space and on bounded domains (Ke Chen, Ruilin Hu and Quoc-Hung Nguyen, 2026)7Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 Weak & singular flows · Same author Yoshikazu Giga · Shared terms “prescribed boundary”, “level set”, “set mean”

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

    Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al.Wider flows PDF

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

  3. Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals

    Hiroyoshi Mitake and Hung V. TranWider flows PDF

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

  4. 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 “level set”

  5. Generic mean curvature flow with obstacles

    Tim Laux and Keisuke TakasaoWider flows PDF

    Topics Weak & singular flows · Shared terms “obstacle”, “level set”

  6. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows PDF

    Topics Weak & singular flows · Shared terms “level set”, “set flow”

  7. Well-posedness for the mean curvature flow on the half-space and on bounded domains

    Ke Chen, Ruilin Hu and Quoc-Hung NguyenWider flows PDF

    Topics Homogeneous spaces, Long-time existence & convergence · Shared terms “Dirichlet”

  8. Generic regularity of Level Set Flows with spherical singularity

    Ao Sun and Jinxin Xue PDF

    Topics Weak & singular flows · Shared terms “level set”, “set flow”

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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.

  3. 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.