The Plateau–Rayleigh Instability of Translating λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-Solitons

被引:0
作者
Antonio Bueno
Rafael López
Irene Ortiz
机构
[1] Centro Universitario de la Defensa de San Javier,Departamento de Ciencias
[2] Universidad de Granada,Departamento de Geometría y Topología
关键词
Translating soliton; stability; minimizer; Plateau–Rayleigh criterion; 53E10; 35C08; 76E17;
D O I
10.1007/s00025-023-02091-2
中图分类号
学科分类号
摘要
Given a unit vector v∈R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf {v}}\in {\mathbb {R}}^3$$\end{document} and λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in {\mathbb {R}}$$\end{document}, a translating λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-soliton is a surface in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^3$$\end{document} whose mean curvature H satisfies H=⟨N,v⟩+λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=\langle N,{\textbf {v}}\rangle +\lambda $$\end{document}, where N is the Gauss map of the surface. In this paper, we extend the phenomenon of instability of Plateau–Rayleigh for translating λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-solitons of cylindrical type, proving that long pieces of these surfaces are unstable. Specifically, we will provide explicit bounds on the length of these unstable surfaces in terms of λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} and the amplitude of the generating curve. It will be also proved that a graphical translating λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}-soliton is a minimizer of the weighted area in a suitable class of surfaces with the same boundary and the same weighted volume.
引用
收藏
相关论文
共 35 条
  • [1] Aguiar D(2002)Stability of rotating liquid films Q. J. Mech. Appl. Math. 55 327-343
  • [2] Bächer C(2021)Rayleigh-Plateau instability of anisotropic interfaces. Part 2. Limited instability of elastic interfaces J. Fluid Mech. 910 A47-219
  • [3] Graessel K(2010)Stability of constrained cylindrical interfaces and the torus lift of Plateau–Rayleigh J. Fluid Mech. 647 201-293
  • [4] Gekle S(2020)Invariant hypersurfaces with linear prescribed mean curvature J. Math. Anal. Appl. 487 281-211
  • [5] Bostwick J(2007)Stability of translating solutions to mean curvature flow Calc. Var. Partial. Differ. Equ. 29 199-215
  • [6] Steen P(1980)The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature Commun. Pure Appl. Math. 33 178-72
  • [7] Bueno A(2003)Isoperimetry of waists and concentration of maps Geom. Funct. Anal. 13 47-1652
  • [8] Ortiz I(2019)Volume growth, entropy and stability for translating solitons Commun. Anal. Geom. 27 1641-48
  • [9] Clutterbuck J(2014)The classification of constant weighted curvature curves in the plane with a log-linear density Commun. Pure Appl. Anal. 13 35-299
  • [10] Schnürer OC(1987)The volume preserving mean curvature flow J. Reine Angew. Math. 382 285-47