\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Stable closed equilibria for anisotropic surface energies: Surfaces with edges

Abstract Related Papers Cited by
  • We study the stability of closed, not necessarily smooth, equilibrium surfaces of an anisotropic surface energy for which the Wulff shape is not necessarily smooth. We show that if the Cahn-Hoffman field can be extended continuously to the whole surface and if the surface is stable, then the surface is, up to rescaling, the Wulff shape.
    Mathematics Subject Classification: Primary: 49Q10; Secondary: 53A015.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z., 185 (1984), 339-353.doi: 10.1007/BF01215045.

    [2]

    J. E. Brothers and F. Morgan, The isoperimetric theorem for general integrands, Michigan Math. J., 41 (1994), 419-431.doi: 10.1307/mmj/1029005070.

    [3]

    J. W. Cahn and D. W. Hoffman, A vector thermodynamics for anisotropic surfaces. II. Curved and faceted surfaces, Acta Metallurgica, 22 (1974), 1205-1214.doi: 10.1016/0001-6160(74)90134-5.

    [4]

    Y. Giga, "Surface Evolution Equations. A Level Set Approach," Monographs in Mathematics, 99, Birkhäuser Verlag, Basel, 2006.

    [5]

    Y. He, H. Li, H. Ma and J. Ge, Compact embedded hypersurfaces with constant higher order anisotropic mean curvatures, Indiana Univ. Math. J., 58 (2009), 853-868.doi: 10.1512/iumj.2009.58.3515.

    [6]

    Y. He and H. Li, A new variational characterization of the Wulff shape, Differential Geom. Appl., 26 (2008), 377-390.

    [7]

    Y. He and H. Li, Stability of hypersurfaces with constant (r+1)-th anisotropic mean curvature, Illinois J. Math., 52 (2008), 1301-1314.

    [8]

    M. Koiso and B. Palmer, Geometry and stability of surfaces with constant anisotropic mean curvature, Indiana Univ. Math. J., 54 (2005), 1817-1852.doi: 10.1512/iumj.2005.54.2613.

    [9]

    M. Koiso and B. Palmer, Stability of anisotropic capillary surfaces between two parallel planes, Calculus of Variations and Partial Differential Equations, 25 (2006), 275-298.

    [10]

    M. Koiso and B. Palmer, Rolling construction for anisotropic Delaunay surfaces, Pacific J. Math., 234 (2008), 345-378.

    [11]

    M. Koiso and B. Palmer, Anisotropic umbilic points and Hopf's theorem for constant anisotropic mean curvature, Indiana Univ. Math. J., 59 (2010), 79-90.doi: 10.1512/iumj.2010.59.4164.

    [12]

    F. Morgan, Planar Wulff shape is unique equilibrium, Proc. Amer. Math. Soc., 133 (2005), 809-813.doi: 10.1090/S0002-9939-04-07661-0.

    [13]

    B. Palmer, Stability of the Wulff shape, Proc. Amer. Math. Soc., 126 (1998), 3661-3667.doi: 10.1090/S0002-9939-98-04641-3.

    [14]

    H. C. Wente, A note on the stability theorem of J. L. Barbosa and M. Do Carmo for closed surfaces of constant mean curvature, Pacific J. Math., 147 (1991), 375-379.

    [15]

    S. Winklmann, A note on the stability of the Wulff shape, Arch. Math. (Basel), 87 (2006), 272-279.doi: 10.1007/s00013-006-1685-y.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(143) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return