Paper · Jun 8, 2020

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

Yoshikazu Giga, Norbert Pozar

math.APWider flowsarXiv:2006.04375
View PDF View HTML arXiv page

As on arXiv

Abstract

A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the value of crystalline curvature may depend not only on the geometry of evolving surfaces but also on the driving force if it is spatially inhomogeneous.

View PDF View HTML arXiv page
Submitted
Jun 8, 2020
Subjects
math.AP

Comments: 23 pages

Mean curvature flowAnisotropic & nonlocal flowsWeak & singular flowsHomogeneous spacesStability & uniqueness

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 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 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)4Požár 2018 On obstacle problem for mean curvature flow with driving force (Yoshikazu Giga, Hung V. Tran and Longjie Zhang, 2018)7Giga 2018 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 A level-set method for a mean curvature flow with a prescribed boundary (Xingzhi Bian, Yoshikazu Giga and Hiroyoshi Mitake, 2023)8Bian 2023
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, Stability & uniqueness · Same authors Yoshikazu Giga, Norbert Pozar · Shared terms “crystalline mean”, “level set”, “viscosity solutions”

  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, Weak & singular flows · Same authors Yoshikazu Giga, Norbert Pozar · 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 Pozar · Shared terms “crystalline”

  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 Pozar · 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, Stability & uniqueness · Shared terms “crystalline mean”, “comparison principle”, “level set”

  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 Pozar · Shared terms “crystalline mean”, “crystalline curvature”

  7. On obstacle problem for mean curvature flow with driving force

    Yoshikazu Giga, Hung V. Tran and Longjie ZhangWider flows

    Topics Mean curvature flow · Same author Yoshikazu Giga · Shared terms “driving force”

  8. A level-set method for a mean curvature flow with a prescribed boundary

    Xingzhi Bian, Yoshikazu Giga and Hiroyoshi MitakeWider flows

    Topics Homogeneous spaces, Weak & singular flows · Same author Yoshikazu Giga · Shared terms “level set”, “set flow”

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

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

  3. 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.