Paper · Apr 1, 2025

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

Takuya Sato

math.APWider flowsarXiv:2504.00452
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Abstract

We consider the free boundary problems of degenerate elliptic equations that describe the level set formulation of the interface motion evolved by anisotropic forced mean curvature flows. The type of free boundary problems in this paper was initially studied as the first-order Hamilton-Jacobi-Isaacs equations arising in pursuit-evasion differential games and applied to the models of first-order front propagation in Soravia (1994). In this paper, we consider an extension of these free boundary problems to the second-order equations and give a deterministic game representation based on a discrete approximation scheme in Kohn and Serfaty (2006). Furthermore, we prove the comparison principle for our free boundary problems by using the framework of time-discrete games.

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Submitted
Apr 1, 2025
Subjects
math.AP

Comments: 32 pages

Mean curvature flowVolume-preserving & constrained flowsWeak & singular flowsFree boundary & capillary

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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)3Jang 2021 Asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals (Hiroyoshi Mitake and Hung V. Tran, 2023)4Mitake 2023 Quenching for axisymmetric hypersurfaces under forced mean curvature flows (Hiroyoshi Mitake, Yusuke Oka and Hung Vinh Tran, 2024)2Mitake 2024 Volume-preserving mean-curvature flow as a singular limit of a diffusion-aggregation equation (Antoine Mellet and Michael Rozowski, 2024)7Mellet 2024 Diffusion-aggregation equations and volume-preserving mean curvature flows (Jiwoong Jang and Antoine Mellet, 2025)8Jang 2025 Parabolic free boundary phase transition and mean curvature flow (Jingeon An-Lacroix and Kiichi Tashiro, 2025)1An-Lacroix 2025 Free boundary flow through cylindrical singularities (Yueheng Bao and Robert Haslhofer, 2026)6Bao 2026 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)5Bao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Parabolic free boundary phase transition and mean curvature flow

    Jingeon An-Lacroix and Kiichi TashiroWider flows

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

  2. Quenching for axisymmetric hypersurfaces under forced mean curvature flows

    Hiroyoshi Mitake, Yusuke Oka and Hung Vinh TranWider flows

    Topics Volume-preserving & constrained flows · Shared terms “forced mean”

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

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

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

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

    Hiroyoshi Mitake and Hung V. TranWider flows

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

  5. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows

    Topics Free boundary & capillary, Weak & singular flows

  6. Free boundary flow through cylindrical singularities

    Yueheng Bao and Robert HaslhoferWider flows

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “level set”

  7. Volume-preserving mean-curvature flow as a singular limit of a diffusion-aggregation equation

    Antoine Mellet and Michael RozowskiWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows · Shared terms “interface”, “elliptic”

  8. Diffusion-aggregation equations and volume-preserving mean curvature flows

    Jiwoong Jang and Antoine MelletWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Weak & singular flows · Shared terms “elliptic”, “formulation”

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) 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) 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. 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.