Paper · Oct 4, 2018

On obstacle problem for mean curvature flow with driving force

Yoshikazu Giga, Hung V. Tran, Longjie Zhang

math.APWider flowsarXiv:1810.03456
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.

View PDF View HTML arXiv page
Submitted
Oct 4, 2018
Subjects
math.AP, math.DS
Mean curvature flowRotational symmetry & warped products

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 On mean curvature flow with driving force starting as singular initial hypersurface (Ryunosuke Mori and Longjie Zhang, 2017)2Mori 2017 Existence of self-similar solution of the inverse mean curvature flow (K. M. Hui, 2018)6Hui 2018 Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms (Yoshikazu Giga, Hiroyoshi Mitake and Hung V. Tran, 2019)1Giga 2019 Viscosity solutions for the crystalline mean curvature flow with a nonuniform driving force term (Yoshikazu Giga and Norbert Pozar, 2020)3Giga 2020 Level-set forced mean curvature flow with the Neumann boundary condition (Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al., 2021)8Jang 2021 Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain (Jiwoong Jang, 2021)7Jang 2021 Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles (Bendong Lou and Lixia Yuan, 2022)4Lou 2022 A level-set method for a mean curvature flow with a prescribed boundary (Xingzhi Bian, Yoshikazu Giga and Hiroyoshi Mitake, 2023)5Bian 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms

    Yoshikazu Giga, Hiroyoshi Mitake and Hung V. TranWider flows

    Topics Mean curvature flow · Same authors Yoshikazu Giga, Hung V. Tran · Shared terms “large time”, “time behavior”

  2. On mean curvature flow with driving force starting as singular initial hypersurface

    Ryunosuke Mori and Longjie ZhangWider flows

    Topics Mean curvature flow · Same author Longjie Zhang · Shared terms “driving force”

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

    Yoshikazu Giga and Norbert PozarWider flows

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

  4. Translating Solutions of a Generalized Mean Curvature Flow in a Cylinder: I. Constant Boundary Angles

    Bendong Lou and Lixia YuanWider flows

    Topics Rotational symmetry & warped products · Shared terms “driving force”, “radially symmetric”

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

    Xingzhi Bian, Yoshikazu Giga and Hiroyoshi MitakeWider flows

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

  6. Existence of self-similar solution of the inverse mean curvature flow

    K. M. HuiWider flows

    Topics Rotational symmetry & warped products · Shared terms “radially symmetric”

  7. Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain

    Jiwoong JangWider flows

    Topics Rotational symmetry & warped products · Shared terms “large time”, “radially symmetric”, “time behavior”

  8. Level-set forced mean curvature flow with the Neumann boundary condition

    Jiwoong Jang, Dohyun Kwon, Hiroyoshi Mitake et al.Wider flows

    Topics Mean curvature flow · Same author Hung V. Tran · Shared terms “large time”, “time behavior”

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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

In the guidebook

  1. 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.