Paper · Feb 10, 2017

Existence and uniqueness for anisotropic and crystalline mean curvature flows

Antonin Chambolle, Massimiliano Morini, Matteo Novaga, Marcello Ponsiglione

math.APWider flowsarXiv:1702.03094
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Abstract

An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a by-product of the analysis, the problem of the convergence of the Almgren-Taylor-Wang minimizing movements scheme to a unique (up to fattening) "flat flow" in the case of general, possibly crystalline, anisotropies is settled.

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Submitted
Feb 10, 2017
Subjects
math.AP, math.NA
Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flowsStability & uniqueness

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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 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)2Giga 2016 Some results on anisotropic fractional mean curvature flows (Antonin Chambolle, Matteo Novaga and Berardo Ruffini, 2016)4Chambolle 2016 Stability results for nonlocal geometric evolutions and limit cases for fractional mean curvature flows (Annalisa Cesaroni, Lucia De Luca, Matteo Novaga et al., 2020)3Cesaroni 2020 Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities (Giovanni Bellettini, Antonin Chambolle and Shokhrukh Yu. Kholmatov, 2020)8Bellettini 2020 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)1Chambolle 2020 Graphical translators for anisotropic and crystalline mean curvature flow (Annalisa Cesaroni, Heiko Kroener and Matteo Novaga, 2021)6Cesaroni 2021 Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2022)7Chambolle 2022 Discrete-to-continuum crystalline curvature flows (Antonin Chambolle, Daniele De Gennaro and Massimiliano Morini, 2024)5Chambolle 2024
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, Stability & uniqueness · Same authors Antonin Chambolle, Matteo Novaga · Shared terms “crystalline mean”, “flat flow”

  2. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows, Stability & uniqueness · Shared terms “crystalline mean”, “solutions satisfy”, “level set”

  3. 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, Stability & uniqueness · Same authors Matteo Novaga, Marcello Ponsiglione · Shared terms “set formulation”, “level set”

  4. Some results on anisotropic fractional mean curvature flows

    Antonin Chambolle, Matteo Novaga and Berardo RuffiniWider flows

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

  5. Discrete-to-continuum crystalline curvature flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows

    Topics Anisotropic & nonlocal flows · Same authors Antonin Chambolle, Massimiliano Morini · Shared terms “crystalline”, “anisotropy”

  6. Graphical translators for anisotropic and crystalline mean curvature flow

    Annalisa Cesaroni, Heiko Kroener and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows, Stability & uniqueness · Same author Matteo Novaga · Shared terms “crystalline mean”, “mobility”, “anisotropy”

  7. Minimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows

    Topics Weak & singular flows · Same authors Antonin Chambolle, Massimiliano Morini · Shared terms “minimizing movements”, “anisotropic”, “movements scheme”

  8. Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

    Giovanni Bellettini, Antonin Chambolle and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Antonin Chambolle · Shared terms “mobility”, “minimizing movements”, “anisotropy”

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.