On the Three-Dimensional Shape of a Crystal

The purpose of this paper is to investigate the Almgren problem in R3 under generic conditions on the potential and tension functions which define the free energy. This problem appears in classical thermodynamics when one seeks to understand whether minimizing the free energy with convex potential in the class of sets of finite perimeter under a mass constraint generates a convex minimizer representing a crystal. Our new idea in proving a three-dimensional convexity theorem is to utilize convexity and a stability theorem when the mass is small, as well as a first-variation partial differential equation along with a new maximum principle approach.