Paper · Oct 22, 2018

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

Yoshikazu Giga, Fumihiko Onoue, Keisuke Takasao

math.APWider flowsarXiv:1810.09107v2
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Abstract

We consider the sharp interface limit of the Allen-Cahn equation with Dirichlet or dynamic boundary conditions and give a varifold characterization of its limit which is formally a mean curvature flow with Dirichlet or dynamic boundary conditions. In order to show the existence of the limit, we apply the phase field method under the vanishing on the boundary and the boundedness of the discrepancy measure. For this purpose, we extend the usual Brakke flow under these boundary conditions by the first variations for varifolds on the boundary.

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Submitted
Oct 22, 2018
Last revised
Dec 28, 2020 (v2)
Subjects
math.AP, math.DG

Comments: 63 pages, 4 figures

Mean curvature flowWeak & singular flowsBoundaries & non-compactPhase fields & multiphase flows

Its neighbourhood

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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 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 Stochastic Allen-Cahn approximation of the mean curvature flow: large deviations upper bound (Lorenzo Bertini, Paolo Buttà and Adriano Pisante, 2016)8Bertini 2016 Convergence of Landau-Lifshitz equation to multi-phase Brakke's mean curvature flow (Keisuke Takasao, 2018)1Takasao 2018 Existence of weak solution for mean curvature flow with transport term and forcing term (Keisuke Takasao, 2019)6Takasao 2019 On obstacle problem for Brakke's mean curvature flow (Keisuke Takasao, 2021)4Takasao 2021 A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)7Hensel 2021 Existence of weak solution to volume preserving mean curvature flow in higher dimensions (Keisuke Takasao, 2022)5Takasao 2022 Brakke inequality and the existence of Brakke-flow for volume preserving mean curvature flow (Andrea Chiesa and Keisuke Takasao, 2025)3Chiesa 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)2Laux 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 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 “varifolds”, “Allen Cahn”, “Cahn equation”

  2. 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 “Allen Cahn”, “interface limit”, “sharp interface”

  3. 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 “varifolds”, “Brakke flows”

  4. On obstacle problem for Brakke's mean curvature flow

    Keisuke TakasaoWider flows

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

  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 “varifolds”, “Cahn equation”, “Allen”

  6. 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 · Same author Keisuke Takasao · Shared terms “Allen Cahn”, “Cahn equation”

  7. A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “varifolds”, “Allen Cahn”, “Cahn equation”

  8. Stochastic Allen-Cahn approximation of the mean curvature flow: large deviations upper bound

    Lorenzo Bertini, Paolo Buttà and Adriano PisanteWider flows

    Topics Phase fields & multiphase flows, Weak & singular flows · Shared terms “Allen Cahn”, “sharp interface”, “Cahn equation”

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) 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) 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.