Paper · Feb 11, 2020

On a Curvature Flow in a Band Domain with Unbounded Boundary Slopes

Lixia Yuan, Wei Zhao

math.APWider flowsarXiv:2002.04549
View PDF View HTML arXiv page

As on arXiv

Abstract

We consider an anisotropic curvature flow in a band domain , where , and denote the unit normal vector, normal velocity and curvature, respectively, of a graphic curve . We consider the case when and the curve contacts with slopes equaling to times of its height (which are unbounded when the solution moves to infinity). First, we present the global well-posedness and then, under some symmetric assumptions on and , we show the uniform interior gradient estimates for the solution. Based on these estimates, we prove that converges as in topology to a cup-like traveling wave with {\it infinite} derivatives on the boundaries.

View PDF View HTML arXiv page
Submitted
Feb 11, 2020
Subjects
math.AP
Other curvature flowsAnisotropic & nonlocal flows

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 Anisotropic curvature flow of immersed curves (Gwenael Mercier, Matteo Novaga and Paola Pozzi, 2016)6Mercier 2016 Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes (Bendong Lou, Xiaoliu Wang and Lixia Yuan, 2019)1Lou 2019 Asymptotic convergence for a class of anisotropic curvature flows (Haizhong Li, Botong Xu and Ruijia Zhang, 2021)5Li 2021 Motion by crystalline-like mean curvature: a survey (Yoshikazu Giga and Norbert Požár, 2021)3Giga 2021 Some aspects of anisotropic curvature flow of planar partitions (Giovanni Bellettini and Shokhrukh Yu. Kholmatov, 2023)2Bellettini 2023 Deforming Locally Convex Curves into Curves of Constant k-order Width (Laiyuan Gao, Horst Martini and Deyan Zhang, 2024)7Gao 2024 Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows (Zhishuai Liu and Guoxin Wei, 2026)4Liu 2026 Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions (Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al., 2026)8Pu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes

    Bendong Lou, Xiaoliu Wang and Lixia YuanWider flows

    Topics Other curvature flows · Same author Lixia Yuan · Shared terms “boundary slopes”, “unbounded boundary”, “band domain”

  2. Some aspects of anisotropic curvature flow of planar partitions

    Giovanni Bellettini and Shokhrukh Yu. KholmatovWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

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

    Yoshikazu Giga and Norbert PožárWider flows

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

  4. Anisotropic-Length-Preserving Weighted Anisotropic Curvature Flows

    Zhishuai Liu and Guoxin WeiWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  5. Asymptotic convergence for a class of anisotropic curvature flows

    Haizhong Li, Botong Xu and Ruijia ZhangWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  6. Anisotropic curvature flow of immersed curves

    Gwenael Mercier, Matteo Novaga and Paola PozziWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  7. Deforming Locally Convex Curves into Curves of Constant -order Width

    Laiyuan Gao, Horst Martini and Deyan ZhangWider flows

    Topics Anisotropic & nonlocal flows, Other curvature flows

  8. Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions

    Tianhong Pu, Xiaoqian Xin, Lixia Yuan et al.Wider flows

    Topics Other curvature flows · Same author Lixia Yuan · Shared terms “normal velocity”

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)