Paper · Dec 10, 2021

Motion by crystalline-like mean curvature: a survey

Yoshikazu Giga, Norbert Požár

math.APWider flowsarXiv:2112.05294
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Abstract

We consider a class of anisotropic curvature flows called a crystalline curvature flow. We present a survey on this class of flows with special emphasis on the well-posedness of its initial value problem.

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Submitted
Dec 10, 2021
Subjects
math.AP

Comments: 76 pages, 11 figures

Other curvature flowsAnisotropic & nonlocal flows

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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)1Giga 2016 Anisotropic curvature flow of immersed curves (Gwenael Mercier, Matteo Novaga and Paola Pozzi, 2016)6Mercier 2016 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)2Giga 2017 On the self-similar solutions of the crystalline mean curvature flow in three dimensions (Norbert Požár, 2018)7Požár 2018 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)8Chambolle 2020 Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term (Yoshikazu Giga and Norbert Pozar, 2020)3Giga 2020 On volume-preserving crystalline mean curvature flow (Inwon Kim, Dohyun Kwon and Norbert Požár, 2020)4Kim 2020 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. A level set crystalline mean curvature flow of surfaces

    Yoshikazu Giga and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows · Same authors Yoshikazu Giga, Norbert Požár · Shared terms “crystalline”

  2. 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 · Same authors Yoshikazu Giga, Norbert Požár · Shared terms “crystalline”, “posedness”

  3. Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term

    Yoshikazu Giga and Norbert PozarWider flows

    Topics Anisotropic & nonlocal flows · Same authors Yoshikazu Giga, Norbert Požár · Shared terms “crystalline”

  4. On volume-preserving crystalline mean curvature flow

    Inwon Kim, Dohyun Kwon and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows · Same author Norbert Požár · Shared terms “crystalline”

  5. Discrete-to-continuum crystalline curvature flows

    Antonin Chambolle, Daniele De Gennaro and Massimiliano MoriniWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows · Shared terms “crystalline”

  6. Anisotropic curvature flow of immersed curves

    Gwenael Mercier, Matteo Novaga and Paola PozziWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows · Shared terms “crystalline”

  7. On the self-similar solutions of the crystalline mean curvature flow in three dimensions

    Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows · Same author Norbert Požár · Shared terms “crystalline”

  8. Anisotropic and crystalline mean curvature flow of mean-convex sets

    Antonin Chambolle and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “crystalline”

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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)