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On the equilibrium shape of a crystal

Emanuel Indrei

Abstract

A solution is given to a long-standing open problem posed by Almgren.

On the equilibrium shape of a crystal

Abstract

A solution is given to a long-standing open problem posed by Almgren.

Paper Structure

This paper contains 9 sections, 30 theorems, 165 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Proof of Theorem \ref{['g']}
  4. A stability theorem
  5. Geometry of $E_m$
  6. Proof of Theorem \ref{['n=2']}
  7. Appendix
  8. Modulus of the free energy
  9. Convexity in higher dimension

Key Result

Theorem 1.1

There exists $g \ge 0$ convex such that $g(0)=0$$\&$ such that if $m>0$, then there is no solution to

Figures (5)

  • Figure 1: $g$
  • Figure 2: $g^\epsilon$
  • Figure 3: (a)
  • Figure 4: A pathological scenario
  • Figure 5: $h'(|x|)=0$ on $B_r(z) \subset E \cap \{g \neq 0\}$, Corollary \ref{['a_radi']}

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4: The radial convex potential
  • Corollary 1.5
  • proof
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • ...and 48 more