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Stability of anisotropic capillary surfaces between two parallel planes. (English) Zbl 1095.76019
The authors extend results of M. Athanassenas [J. Reine Angew. Math. 377, 97–107 (1987; Zbl 0604.53003)] and Th. I. Vogel [SIAM J. Appl. Math. 47, 516–525 (1987; Zbl 0627.53004)] on the stability of capillary surfaces between two parallel planes to anisotropic rotational symmetric surface energies. It is shown that any stable embedded equilibrium capillary surface between two horizontal planes with nonempty free boundary on these planes is either a sufficiently short cylinder or a suitable part of the Wulff shape which is a hemisphere in the case of a homogeneous energy functional, see Athanassenas or Vogel. The proof is based on previous studies of the authors into anisotropic Delaunay surfaces, a generalization of Wente’s (1966) second variation formula and on explicit calculations similar to Athanassenas and Vogel.
MSC:
76D45Capillarity (surface tension)
76M30Variational methods (fluid mechanics)
References:
[1]Athanassenas, M.: A variational problem for constant mean curvature surfaces with free boundary. J. Reine Angew. Math. 377, 97–107 (1987) · Zbl 0604.53003 · doi:10.1515/crll.1987.377.97
[2]Barbosa, J.L., Do Carmo, M., Eschenburg, J.: Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math. Zeit. 197, 123–138 (1988) · Zbl 0653.53045 · doi:10.1007/BF01161634
[3]Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. 1. John Wiley, New York (1989)
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[5]Federer, H.: Geometric Measure Theory, Die Grundlehren der Mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969)
[6]Finn, R.: Equilibrium Capillary Surfaces, Grundlehren der Mathematischen Wissenschaften, 284. Springer-Verlag, New York (1986)
[7]Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics, Springer-Verlag, Berlin (2001)
[8]Hopf, H.: Differential Geometry in the Large, Second edition, Lecture Notes in Mathematics 1000. Springer-Verlag, Berlin (1989)
[9]Koiso, M., Palmer, B.: Geometry and stability of surfaces with constant anisotropic mean curvature preprint.
[10]Reilly, Robert C.: The relative differential geometry of nonparametric hypersurfaces. Duke Math. J. 43, 705–721 (1976) · Zbl 0342.53015 · doi:10.1215/S0012-7094-76-04355-6
[11]Palmer, B.: Stability of the Wulff shape. Proc. Amer. Math. Soc. 126, 3661–3667 (1998) · Zbl 0924.53009 · doi:10.1090/S0002-9939-98-04641-3
[12]Taylor, J.E.: Crystalline variational problems. Bull. Amer. Math. Soc. 84, 568–588 (1978) · Zbl 0392.49022 · doi:10.1090/S0002-9904-1978-14499-1
[13]Vogel, T.I.: Stability of a liquid drop trapped between two parallel planes. SIAM J. Appl. Math. 47, 516–525 (1987) · Zbl 0627.53004 · doi:10.1137/0147034