Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem

• Main Author: Miyuki Koiso
• Format: Article
• Language: English
• Published: MDPI AG 2019-10-01
• Series: Mathematical and Computational Applications
• Subjects: anisotropic mean curvature; anisotropic surface energy; wulff shape; anisotropic mean curvature flow; crystalline variational problem; capillary problem
• Online Access: https://www.mdpi.com/2297-8747/24/4/88

We study a variational problem for hypersurfaces in the Euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered hypersurface, which was introduced to model the surface tension of a small crystal. The purpose of this paper is two-fold. First, we give uniqueness and nonuniqueness results for closed equilibria under weaker assumptions on the regularity of both considered hypersurfaces and the anisotropic surface energy density than previous works and apply the results to the anisotropic mean curvature flow. This part is an announcement of two forthcoming papers by the author. Second, we give a new uniqueness result for stable anisotropic capillary surfaces in a wedge in the three-dimensional Euclidean space.