Paper · Feb 8, 2026

Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions

Linlin Fan, Peibiao Zhao

math.DGWider flowsarXiv:2602.07989
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Abstract

Wang, Weng and Xia[Math. Ann. 388 (2024), no. 2] studied a mean curvature type flow for the smooth, embedded capillary hypersurfaces with a constant contact angle and confirmed the existence of solutions by the standard PDE theory. In the present paper, we study a fractional mean curvature flow for -regular hypersurfaces with a capillary-type boundary condition and obtain the short time existence by the fixed point argument.

Mean curvature flowVolume-preserving & constrained flowsAnisotropic & nonlocal flowsBoundaries & non-compactFree boundary & capillaryHarmonic maps & DeTurck trick

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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 Mean curvature type flow and sharp Micheal-Simon inequalities (J. Cui and P. Zhao, 2021)4Cui 2021 Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain (Jiwoong Jang, 2021)1Jang 2021 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)6Cesaroni 2022 Short time existence and smoothness of the nonlocal mean curvature flow of graphs (Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias Weth, 2022)3Attiogbe 2022 A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space (Yingxiang Hu, Yong Wei, Bo Yang et al., 2022)7Hu 2022 Convergence of the volume preserving fractional mean curvature flow for convex sets (Vesa Julin and Domenico Angelo La Manna, 2023)8Julin 2023 Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions (Can Cui and Nung Kwan Yip, 2025)2Cui 2025 Two dimensional anisotropic mean curvature flow with contact angle condition (Can Cui and Nung Kwan Yip, 2025)5Cui 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Shared terms “capillary type”, “type boundary”, “boundary condition”

  2. Anisotropic mean curvature flow with contact angle and Neumann boundary conditions in arbitrary dimensions

    Can Cui and Nung Kwan YipWider flows PDF

    Topics Anisotropic & nonlocal flows, Free boundary & capillary, Boundaries & non-compact · Shared terms “boundary condition”, “contact angle”

  3. Short time existence and smoothness of the nonlocal mean curvature flow of graphs

    Anoumou Attiogbe, Mouahmed Moustapha Fall and Tobias WethWider flows PDF

    Topics Anisotropic & nonlocal flows, Harmonic maps & DeTurck trick · Shared terms “fractional mean”, “classical solution”, “short time”

  4. Mean curvature type flow and sharp Micheal-Simon inequalities

    J. Cui and P. ZhaoWider flows PDF

    Topics Volume-preserving & constrained flows · Same author Peibiao Zhao · Shared terms “type flow”, “curvature type”

  5. Two dimensional anisotropic mean curvature flow with contact angle condition

    Can Cui and Nung Kwan YipWider flows PDF

    Topics Anisotropic & nonlocal flows, Free boundary & capillary, Boundaries & non-compact · Shared terms “boundary condition”, “contact angle”

  6. Stability of the ball under volume preserving fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows PDF

    Topics Anisotropic & nonlocal flows · Shared terms “fractional mean”, “time existence”

  7. A complete family of Alexandrov-Fenchel inequalities for convex capillary hypersurfaces in the half-space

    Yingxiang Hu, Yong Wei, Bo Yang et al.Wider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Boundaries & non-compact · Shared terms “capillary hypersurfaces”, “contact angle”, “boundary condition”

  8. Convergence of the volume preserving fractional mean curvature flow for convex sets

    Vesa Julin and Domenico Angelo La MannaWider flows PDF

    Topics Anisotropic & nonlocal flows · Shared terms “fractional 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. 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) Anisotropic & nonlocal flows Wider flows When surface energy depends on direction, as in crystals, mean curvature flow becomes anisotropic, and the Wulff shape replaces the round sphere. Start with: Crystalline variational problems (Taylor, 1978) 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. 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.