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Anisotropic capillary surfaces with wetting energy. (English) Zbl 1136.76011
The authors generalize results of T. I. Vogel and R. Finn [Z. Anal. Anwend. 11, No. 1, 3–23 (1992; Zbl 0760.76015)] and M. Athanassenas [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 26, No. 4, 749–762 (1998; Zbl 0927.76011)] concerning liquid drops between two parallel planes to free anisotropic energies of the type $ℱ\left[X\right]:={\int }_{{\Sigma }}F\left(\nu \right)\phantom{\rule{4pt}{0ex}}d{\Sigma }$, where $\nu =\left({\nu }_{1},{\nu }_{2},{\nu }_{3}\right)$: ${\Sigma }\to {S}^{2}$ is the Gauss map of $X$: ${\Sigma }\to {ℝ}^{3}$. The reason for this generalization is that the classical isotropic surface tension must be replaced by an anisotropic one if the materials involved enter a solid or liquid crystal phase. In particular, the paper contains a derivation of the first and second variation of rotationally symmetric anisotropic surface energies and a stability analysis of anisotropic Delaunay surfaces which are equilibria of the problem. The stability analysis is based on the study of eigenvalues of a linear second-order ordinary differential equation associated to the second variation.
##### MSC:
 76B45 Capillarity (surface tension) 76E17 Interfacial stability and instability (fluid dynamics) 49Q05 Minimal surfaces (calculus of variations)
##### Keywords:
stability analysis; anisotropic Delaunay surfaces
##### References:
 [1] Athanassenas M. (1987). A variational problem for constant mean curvature surfaces with free boundary. J. Reine Angew. Math. 377: 97–107 · Zbl 0604.53003 · doi:10.1515/crll.1987.377.97 [2] Barbosa J.L. and Do Carmo M. (1984). Stability of hypersurfaces with constant mean curvature. Math. Z. 185: 339–353 · Zbl 0529.53006 · doi:10.1007/BF01215045 [3] Brothers J.E. and Morgan F. (1994). The isoperimetric theorem for general integrands. Michigan Math. J. 41: 419–431 · Zbl 0923.49019 · doi:10.1307/mmj/1029005070 [4] Courant, R., Hilbert, D.: Methods of Mathematical Physics, Vol. 1, Interscience Publishers, Inc., New York, N.Y., 1953 [5] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Grundlehren der mathematischen Wissenschaften 224. Springer, Berlin Heidelberg New York (1983) [6] Hopf, H.: Differential geometry in the Large, 2nd edn. Lecture Notes in Mathematics, vol. 1000, Springer, Berlin Heidelberg New York (1989) [7] Koiso M. (1986). Symmetry of hypersurfaces of constant mean curvature with symmetric boundary. Math. Zeit. 191: 567–574 · Zbl 0563.53007 · doi:10.1007/BF01162346 [8] Koiso M. (2002). Deformation and stability of surfaces with constant mean curvature. Tohoku Math. J. (2) 54: 145–159 · Zbl 1010.58007 · doi:10.2748/tmj/1113247184 [9] Koiso M. and Palmer B. (2005). On a variational problem for soap films with gravity and partially free boundary. J. Math. Soc. Jpn. 57: 333–355 · Zbl 1072.49029 · doi:10.2969/jmsj/1158242062 [10] Koiso M. and Palmer B. (2005). Geometry and stability of surfaces with constant anisotropic mean curvature. Ind. Univ. Math. J. 54: 1817–1852 · Zbl 1120.58010 · doi:10.1512/iumj.2005.54.2613 [11] Koiso M. and Palmer B. (2006). Stability of anisotropic capillary surfaces between two parallel planes. Calcul. Vari. Partial Differential Equations 25: 275–298 · Zbl 1095.76019 · doi:10.1007/s00526-005-0336-7 [12] Vogel T.I. (1987). Stability of a liquid drop trapped between two parallel planes. SIAM J. Appl. Math. 47: 516–525 · Zbl 0627.53004 · doi:10.1137/0147034 [13] Vogel T.I. (1989). Stability of a liquid drop trapped between two parallel planes II, general contact angles. SIAM J. Appl. Math. 49: 1009–1028 · Zbl 0691.53007 · doi:10.1137/0149061 [14] Winterbottom W.L. (1967). Equilibrium shape of a small particle in contact with a foreign substrate. Acta Metal. 15: 303–310 · doi:10.1016/0001-6160(67)90206-4 [15] Zhou L. (1997). On stability of a catenoidal liquid bridge. Pacific J. Math. 178: 185–198 · Zbl 0868.76019 · doi:10.2140/pjm.1997.178.185