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The one-dimensional equilibrium shape of a crystal

Emanuel Indrei

TL;DR

The paper addresses the Almgren-type question of whether minimizing the one-dimensional free energy $\mathcal{E}(E)=\mathcal{F}(E)+\mathcal{G}(E)$ under a mass constraint $|E|=m$ yields convex minimizers when the sublevel sets of $g$ are convex and $g(0)=0$, $g\ge0$. It proves that in one dimension minimizers are intervals (up to translation) by reducing the problem to translates of the fixed interval $I=(0,m)$ and deriving a key inequality (Claim 2) via a simple rearrangement or optimal transport argument. The minimizer structure is characterized as $E_m = I_a+\alpha$, with the translation $\alpha$ determined by the stationarity condition $g(\alpha+m)=g(\alpha)$, and the endpoint configuration determined by the behavior of $g$ on the halves of the line. This 1D result clarifies the Almgren problem in a tractable setting and offers a basis for understanding higher-dimensional behavior via strip limits.

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.

The one-dimensional equilibrium shape of a crystal

TL;DR

The paper addresses the Almgren-type question of whether minimizing the one-dimensional free energy under a mass constraint yields convex minimizers when the sublevel sets of are convex and , . It proves that in one dimension minimizers are intervals (up to translation) by reducing the problem to translates of the fixed interval and deriving a key inequality (Claim 2) via a simple rearrangement or optimal transport argument. The minimizer structure is characterized as , with the translation determined by the stationarity condition , and the endpoint configuration determined by the behavior of on the halves of the line. This 1D result clarifies the Almgren problem in a tractable setting and offers a basis for understanding higher-dimensional behavior via strip limits.

Abstract

Optimizing the free energy under a mass constraint may generate a convex crystal subject to assumptions on the potential , . 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.
Paper Structure (5 sections, 1 theorem, 74 equations)

This paper contains 5 sections, 1 theorem, 74 equations.

Table of Contents

  1. Introduction
  2. The proof
  3. Proof of Theorem \ref{['n=1']}
  4. Optimal transport proof
  5. Identifying $\alpha$

Key Result

Theorem 1.1

If $n=1$, $m \in (0, \infty)$, $g(0)=0$, $g \ge 0$, and the sub-level sets $\{g < t\}$ are convex, then admits minimizers and any minimizer $E_m$ is convex.

Theorems & Definitions (4)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4