The one-dimensional equilibrium shape of a crystal

TL;DR

This paper proves that in one dimension, the equilibrium shape of a crystal minimizing free energy under a mass constraint is convex, confirming classical assumptions for this specific case.

Contribution

It establishes a rigorous proof that the equilibrium shape of a crystal in one dimension is convex under certain potential conditions, addressing a classical problem.

Findings

The equilibrium shape is convex in one dimension.

The proof confirms classical assumptions for convexity.

The result applies to potentials with g(0)=0 and g ≥ 0.

Abstract

Optimizing the free energy under a mass constraint may generate a convex crystal subject to assumptions on the potential $g(0)=0$, $g \ge 0$. The general problem classically attributed to Almgren is to infer if this is the case assuming the sub-level sets of g are convex. The theorem proven in the paper is that in one dimension the answer is positive.