Stable closed equilibria for anisotropic surface energies: Surfaces with edges

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. In this paper, we will study the stability of closed surfaces which are in equilibrium for an anisotropic surface energy. Neither the equilibrium surface Σ nor the Wulff shape W are assumed to be smooth; they may have edges as depicted in Figure 1. The equilibrium conditions for Σ are expressed by the anisotropic mean curvature being constant on each face of Σ and at each edge, the force balancing condition found by Cahn and Hoffman [3] is satisfied. The main result states that if the the Cahn-Hoffman field extends to a continuous map of Σ to W and if the surface is stable, then the surface agrees with W up to rescaling. Stability means that the second variation of energy is non negative for those variations preserving the volume enclosed by Σ. The condition of extendability of the Cahn-Hoffman map seems natural to us since it implies the force balancing condition along the edges and we can produce many (non closed) examples for which it holds. The first theorem of the type described above is due to Barbosa-do Carmo, [1] who showed that the spheres are the unique stable closed constant mean curvature hypersurfaces in Euclidean space. Later, Wente [12] greatly simplified the proof of this result. Wente’s idea was used by the author [11] to extend the results of [1] to the anisotropic case, assuming both the smoothness of the hypersurface and the smoothness of the Wulff shape. A proof of this result more in the style of [1] was given in [13]. Similar results involving higher order anisotropic mean curvatures can be found in [7]. Because of the applications of anisotropic surface energies to studying inter-faces of structured materials, we have decided to limit our attention to the case of two dimensional surfaces in three space. It is very likely that these ideas can be extended to higher dimensions. In addition, we have limited our discussion for the most part to embedded surfaces which is not essential. 1 a