Paper · Feb 25, 2021

Convergence of supercritical fractional flows to the mean curvature flow

Lucia De Luca, Andrea Kubin, Marcello Ponsiglione

math.APWider flowsarXiv:2102.13041
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Abstract

We consider a core-radius approach to nonlocal perimeters governed by isotropic kernels having critical and supercritical exponents, extending the nowadays classical notion of -fractional perimeter, defined for , to the case \,. We show that, as the core-radius vanishes, such core-radius regularized -fractional perimeters, suitably scaled, -converge to the standard Euclidean perimeter. Under the same scaling, the first variation of such nonlocal perimeters gives back regularized -fractional curvatures which, as the core radius vanishes, converge to the standard mean curvature; as a consequence, we show that the level set solutions to the corresponding nonlocal geometric flows, suitably reparametrized in time, converge to the standard mean curvature flow. Finally, we prove analogous results in the case of anisotropic kernels with applications to dislocation dynamics. Keywords: Fractional perimeters; -convergence; Local and nonlocal geometric evolutions; Viscosity solutions; Level set formulation; Fractional mean curvature flow; Dislocation dynamics

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Submitted
Feb 25, 2021
Subjects
math.AP
Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flows

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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 A level set crystalline mean curvature flow of surfaces (Yoshikazu Giga and Norbert Požár, 2016)4Giga 2016 Existence and uniqueness for anisotropic and crystalline mean curvature flows (Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al., 2017)2Chambolle 2017 Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow (Yoshikazu Giga and Norbert Požár, 2017)8Giga 2017 Eventual Regularization of Fractional Mean Curvature Flow (Stephen Cameron, 2019)6Cameron 2019 Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows (Annalisa Cesaroni, Lucia De Luca, Matteo Novaga et al., 2020)1Cesaroni 2020 Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem (De Gennaro Daniele, Andrea Kubin and Anna Kubin, 2022)3Daniele 2022 A mean curvature flow arising in adversarial training (Leon Bungert, Tim Laux and Kerrek Stinson, 2024)5Bungert 2024 A variational approach to the volume-preserving anisotropic mean curvature flow in 2D (Andrea Kubin, Domenico Angelo La Manna and Enrico Pasqualetto, 2025)7Kubin 2025
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  1. 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, Weak & singular flows · Same authors Lucia De Luca, Marcello Ponsiglione · Shared terms “nonlocal geometric”, “fractional”, “geometric evolution”

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

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

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Marcello Ponsiglione · Shared terms “regularized”, “set formulation”, “suitably”

  3. Asymptotic of the Discrete Volume-Preserving Fractional Mean Curvature Flow via a Nonlocal Quantitative Alexandrov Theorem

    De Gennaro Daniele, Andrea Kubin and Anna KubinWider flows

    Topics Anisotropic & nonlocal flows · Same author Andrea Kubin · Shared terms “fractional”, “perimeter”

  4. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “level set”, “regularized”, “set formulation”

  5. A mean curvature flow arising in adversarial training

    Leon Bungert, Tim Laux and Kerrek StinsonWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “nonlocal perimeter”, “geometric evolution”, “variation”

  6. Eventual Regularization of Fractional Mean Curvature Flow

    Stephen CameronWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “fractional”

  7. A variational approach to the volume-preserving anisotropic mean curvature flow in 2D

    Andrea Kubin, Domenico Angelo La Manna and Enrico PasqualettoWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Andrea Kubin

  8. Approximation of general facets by regular facets with respect to anisotropic total variation energies and its application to the crystalline mean curvature flow

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Shared terms “viscosity solutions”, “set formulation”, “level 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) 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)

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.