Paper · Feb 17, 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, Norbert Požár

math.APWider flowsarXiv:1702.05220
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Abstract

We show that every bounded subset of an Euclidean space can be approximated by a set that admits a certain vector field, the so-called Cahn-Hoffman vector field, that is subordinate to a given anisotropic metric and has a square-integrable divergence. More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner. We use this approximation to construct test functions necessary to prove the comparison principle for viscosity solutions of the level set formulation of the crystalline mean curvature flow that were recently introduced by the authors. As a consequence, we obtain the well-posedness of the viscosity solutions in an arbitrary dimension, which extends the validity of the result in the previous paper.

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Submitted
Feb 17, 2017
Subjects
math.AP
Mean curvature flowAnisotropic & nonlocal flowsWeak & singular 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 Existence and uniqueness for anisotropic and crystalline mean curvature flows (Antonin Chambolle, Massimiliano Morini, Matteo Novaga et al., 2017)5Chambolle 2017 Motion of level sets by inverse anisotropic mean curvature (Francesco Della Pietra, Nunzia Gavitone and Chao Xia, 2018)8Della Pietra 2018 On the self-similar solutions of the crystalline mean curvature flow in three dimensions (Norbert Požár, 2018)4Požár 2018 Anisotropic and crystalline mean curvature flow of mean-convex sets (Antonin Chambolle and Matteo Novaga, 2020)7Chambolle 2020 Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term (Yoshikazu Giga and Norbert Pozar, 2020)2Giga 2020 On volume-preserving crystalline mean curvature flow (Inwon Kim, Dohyun Kwon and Norbert Požár, 2020)6Kim 2020 Motion by crystalline-like mean curvature: a survey (Yoshikazu Giga and Norbert Požár, 2021)3Giga 2021
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, Weak & singular flows · Same authors Yoshikazu Giga, Norbert Požár · Shared terms “crystalline mean”, “viscosity solutions”, “level set”

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

    Yoshikazu Giga and Norbert PozarWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same authors Yoshikazu Giga, Norbert Požár · Shared terms “crystalline mean”, “viscosity solutions”, “comparison principle”

  3. Motion by crystalline-like mean curvature: a survey

    Yoshikazu Giga and Norbert PožárWider flows

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

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

    Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”, “level set”

  5. 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 · Shared terms “crystalline mean”, “set formulation”, “comparison principle”

  6. On volume-preserving crystalline mean curvature flow

    Inwon Kim, Dohyun Kwon and Norbert PožárWider flows

    Topics Anisotropic & nonlocal flows, Weak & singular flows · Same author Norbert Požár · Shared terms “crystalline mean”

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

    Antonin Chambolle and Matteo NovagaWider flows

    Topics Anisotropic & nonlocal flows · Shared terms “crystalline mean”

  8. Motion of level sets by inverse anisotropic mean curvature

    Francesco Della Pietra, Nunzia Gavitone and Chao XiaWider flows

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

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

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    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.10
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