# American Institute of Mathematical Sciences

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

## Stable closed equilibria for anisotropic surface energies: Surfaces with edges

 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.
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. doi: 10.1007/BF01215045. [2] J. E. Brothers and F. Morgan, The isoperimetric theorem for general integrands,, Michigan Math. J., 41 (1994), 419. 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. doi: 10.1016/0001-6160(74)90134-5. [4] Y. Giga, "Surface Evolution Equations. A Level Set Approach,", Monographs in Mathematics, 99 (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. 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. [7] Y. He and H. Li, Stability of hypersurfaces with constant (r+1)-th anisotropic mean curvature,, Illinois J. Math., 52 (2008), 1301. [8] M. Koiso and B. Palmer, Geometry and stability of surfaces with constant anisotropic mean curvature,, Indiana Univ. Math. J., 54 (2005), 1817. 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. [10] M. Koiso and B. Palmer, Rolling construction for anisotropic Delaunay surfaces,, Pacific J. Math., 234 (2008), 345. [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. doi: 10.1512/iumj.2010.59.4164. [12] F. Morgan, Planar Wulff shape is unique equilibrium,, Proc. Amer. Math. Soc., 133 (2005), 809. doi: 10.1090/S0002-9939-04-07661-0. [13] B. Palmer, Stability of the Wulff shape,, Proc. Amer. Math. Soc., 126 (1998), 3661. 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. [15] S. Winklmann, A note on the stability of the Wulff shape,, Arch. Math. (Basel), 87 (2006), 272. 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. doi: 10.1007/BF01215045. [2] J. E. Brothers and F. Morgan, The isoperimetric theorem for general integrands,, Michigan Math. J., 41 (1994), 419. 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. doi: 10.1016/0001-6160(74)90134-5. [4] Y. Giga, "Surface Evolution Equations. A Level Set Approach,", Monographs in Mathematics, 99 (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. 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. [7] Y. He and H. Li, Stability of hypersurfaces with constant (r+1)-th anisotropic mean curvature,, Illinois J. Math., 52 (2008), 1301. [8] M. Koiso and B. Palmer, Geometry and stability of surfaces with constant anisotropic mean curvature,, Indiana Univ. Math. J., 54 (2005), 1817. 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. [10] M. Koiso and B. Palmer, Rolling construction for anisotropic Delaunay surfaces,, Pacific J. Math., 234 (2008), 345. [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. doi: 10.1512/iumj.2010.59.4164. [12] F. Morgan, Planar Wulff shape is unique equilibrium,, Proc. Amer. Math. Soc., 133 (2005), 809. doi: 10.1090/S0002-9939-04-07661-0. [13] B. Palmer, Stability of the Wulff shape,, Proc. Amer. Math. Soc., 126 (1998), 3661. 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. [15] S. Winklmann, A note on the stability of the Wulff shape,, Arch. Math. (Basel), 87 (2006), 272. doi: 10.1007/s00013-006-1685-y.
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