Equilibria for anisotropic surface energies with wetting and line tension

We study the stability of surfaces trapped between two parallel planes with free boundary on these planes. The energy functional consists of anisotropic surface energy, wetting energy, and line tension. Equilibrium surfaces are surfaces with constant anisotropic mean curvature. We study the case where the Wulff shape is of a special type of “product form”, that is, its horizontal sections are all homothetic and has a certain symmetry. Such an anisotropic surface energy is a natural generalization of the area of the surface. Especially, we study the stability of parts of anisotropic Delaunay surfaces which arise as equilibrium surfaces. They are surfaces of the same product form of the Wulff shape. We show that, for these surfaces, the stability analysis can be reduced to the case where the surface is axially symmetric and the functional is replaced by an appropriate axially symmetric one. Moreover, The first author is partially supported by Grant-in-Aid for Scientific Research (C) No. 19540217 and Grant-in-Aid for challenging Exploratory Research No. 22654009 of the Japan Society for the Promotion of Science. The second author was partially funded by Fellowship S-08154 from the Japan Society for the Promotion of Science and by Fundación Séneca project 04540/GERM/06, Spain. Miyuki Koiso Faculty of Mathematics Kyushu University & PRESTO, JST 744 Motooka, Nishi-ku Fukuoka 819-0395, Japan Tel.: +81-92-802-4406 Fax: +81-92-802-4405 E-mail: [email protected] Bennett Palmer Department of Mathematics Idaho State University Pocatello, ID 83209 U.S.A. Tel.: +1-208-282-2402 Fax: +1-208-282-2636 E-mail: [email protected] 2 Miyuki Koiso, Bennett Palmer we obtain necessary and sufficient conditions for the stability of anisotropic sessile drops.