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March  2012, 4(1): 89-97. doi: 10.3934/jgm.2012.4.89

Stable closed equilibria for anisotropic surface energies: Surfaces with edges

Bennett Palmer 1,

1. 

Department of Mathematics, Idaho State University, Pocatello , Idaho, 83209, United States

Received  November 2011 Revised  April 2012 Published  April 2012

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.
Keywords: Anisotropic mean curvature, second variation..
Mathematics Subject Classification: Primary: 49Q10; Secondary: 53A01.
Citation: Bennett Palmer. Stable closed equilibria for anisotropic surface energies: Surfaces with edges. Journal of Geometric Mechanics, 2012, 4 (1) : 89-97. doi: 10.3934/jgm.2012.4.89
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

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