Paper · Feb 12, 2021

On obstacle problem for Brakke's mean curvature flow

Keisuke Takasao

math.APWider flowsarXiv:2102.06375v3
View PDF View HTML arXiv page

As on arXiv

Abstract

We consider the obstacle problem of the weak solution for the mean curvature flow, in the sense of Brakke's mean curvature flow. We prove the global existence of the weak solution with obstacles which have boundaries, in two and three space dimensions. To obtain the weak solution, we use the Allen-Cahn equation with forcing term.

View PDF View HTML arXiv page
Submitted
Feb 12, 2021
Last revised
Aug 27, 2021 (v3)
Subjects
math.AP

Comments: 14 pages

Mean curvature flowWeak & singular flowsLong-time existence & convergencePhase fields & multiphase flows

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 2016, 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 Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow (Keisuke Takasao, 2018)2Takasao 2018 A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions (Yoshikazu Giga, Fumihiko Onoue and Keisuke Takasao, 2018)6Giga 2018 Existence of weak solution for mean curvature flow with transport term and forcing term (Keisuke Takasao, 2019)1Takasao 2019 Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)5Takasao 2022 New advances on the existence and regularity of Brakke's mean curvature flows (Salvatore Stuvard, 2023)8Stuvard 2023 Existence of weak solutions to volume-preserving mean curvature flow with obstacles (Jiwoong Jang, 2025)4Jang 2025 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)7Chiesa 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)3Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence of weak solution for mean curvature flow with transport term and forcing term

    Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows, Long-time existence & convergence · Same author Keisuke Takasao · Shared terms “forcing term”, “Allen Cahn”, “Cahn equation”

  2. Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow

    Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Brakke mean”, “Allen Cahn”, “Cahn equation”

  3. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

    Tim Laux and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Brakke”, “Allen Cahn”, “Cahn equation”

  4. Existence of weak solutions to volume-preserving mean curvature flow with obstacles

    Jiwoong JangWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows, Long-time existence & convergence · Shared terms “obstacle”, “forcing term”, “Allen Cahn”

  5. Existence of weak solution to volume preserving mean curvature flow in higher dimensions

    Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Cahn equation”, “Allen”

  6. A varifold formulation of mean curvature flow with Dirichlet or dynamic boundary conditions

    Yoshikazu Giga, Fumihiko Onoue and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Brakke”, “Allen Cahn”, “Cahn equation”

  7. Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow

    Andrea Chiesa and Keisuke TakasaoWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Same author Keisuke Takasao · Shared terms “Brakke”

  8. New advances on the existence and regularity of Brakke's mean curvature flows

    Salvatore StuvardWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows, Long-time existence & convergence · Shared terms “Brakke mean”

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 in the L² sense. 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) 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) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)

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.