Uniqueness for closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature (Analysis on Shapes of Solutions to Partial Differential Equations)

Abstract

We discuss a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional euclidean space. The energy functional is an anisotropic energy which is a natural generalization of the area for surfaces. However, equilibrium hypersurfaces of this energy are not smooth in general. Locally they are solutions of a second order quasilinear PDE, which is elliptic in a part, and hyperbolic in the rest. In this article, we study the uniqueness problem for closed embedded equilibrium hypersurfaces and give an application to the anisotropic mean curvature flow. This article plays a role of an announcement of a part of the forthcoming paper [6].