On the three-dimensional shape of a crystal

TL;DR

This paper resolves the Almgren problem in three dimensions under certain conditions, demonstrating convexity of crystals formed by free energy minimization, using stability, convexity, and PDE techniques.

Contribution

It introduces a novel approach combining stability theorems, convexity, and PDE methods to settle the Almgren problem in 3D.

Findings

Proves convexity of crystals under specific conditions

Develops a new maximum principle for PDE analysis

Establishes a complete solution to the Almgren problem in 3D

Abstract

In this paper we completely settle the Almgren problem in $\mathbb R^3$ under some generic conditions on the potential and tension functions. The problem, among other things, appears in classical thermodynamics when one is to understand if minimizing the free energy with convex potential and under a mass constraint generates a convex crystal. Our new idea in proving a three-dimensional convexity theorem is to utilize a stability theorem when $m$ is small, convexity when $m$ is small, and the first variation PDE with a new maximum principle approach.