The one-dimensional equilibrium shape of a crystal

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.