Paper · Mar 23, 2016

Some results on anisotropic fractional mean curvature flows

Antonin Chambolle, Matteo Novaga, Berardo Ruffini

math.NAWider flowsarXiv:1603.07239
View PDF View HTML arXiv page

As on arXiv

Abstract

We show the consistency of a threshold dynamics type algorithm for the anisotropic motion by fractional mean curvature, in the presence of a time dependent forcing term. Beside the consistency result, we show that convex sets remain convex during the evolution, and the evolution of a bounded convex set is uniquely defined.

View PDF View HTML arXiv page
Submitted
Mar 23, 2016
Subjects
math.NA, math.AP
Mean curvature flowAnisotropic & nonlocal 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 Existence and uniqueness for anisotropic and crystalline mean curvature flows (Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al., 2017)3Chambolle 2017 Convex sets evolving by volume preserving fractional mean curvature flows (Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci, 2018)8Cinti 2018 Symmetric self-shrinkers for the fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2018)6Cesaroni 2018 Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows (Annalisa Cesaroni, Lucia De Luca, Matteo Novaga et al., 2020)7Cesaroni 2020 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)1Chambolle 2020 Fractional mean curvature flow of Lipschitz graphs (Annalisa Cesaroni and Matteo Novaga, 2021)4Cesaroni 2021 Stability of the ball under volume preserving fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2022)5Cesaroni 2022 Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term (Bohdan Bulanyi and Berardo Ruffini, 2025)2Bulanyi 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Anisotropic and crystalline mean curvature flow of mean-convex sets

    Antonin Chambolle and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Same authors Antonin Chambolle, Matteo Novaga · Shared terms “convex set”

  2. Threshold dynamics approximation schemes for anisotropic mean curvature flows with a forcing term

    Bohdan Bulanyi and Berardo RuffiniWider flows

    Topics Anisotropic & nonlocal flows · Same author Berardo Ruffini · Shared terms “threshold dynamics”, “forcing term”, “dynamics type”

  3. Existence and uniqueness for anisotropic and crystalline mean curvature flows

    Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows · Same authors Antonin Chambolle, Matteo Novaga · Shared terms “forcing term”, “time dependent”, “presence”

  4. Fractional mean curvature flow of Lipschitz graphs

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Same author Matteo Novaga · Shared terms “fractional mean”

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

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Same author Matteo Novaga · Shared terms “fractional mean”

  6. Symmetric self-shrinkers for the fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Same author Matteo Novaga · Shared terms “fractional mean”

  7. Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows

    Annalisa Cesaroni, Lucia De Luca, Matteo Novaga et al.Wider flows

    Topics Anisotropic & nonlocal flows · Same author Matteo Novaga · Shared terms “fractional mean”

  8. Convex sets evolving by volume preserving fractional mean curvature flows

    Eleonora Cinti, Carlo Sinestrari and Enrico ValdinociWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “fractional mean”, “convex set”

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) 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)

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.