Stable closed equilibria for anisotropic surface energies: Surfaces with edges

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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

Abstract
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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.
[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.

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.
[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.

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