Strong stability for the Wulff inequality with a crystalline norm

TL;DR

This paper establishes a sharp stability inequality for the Wulff energy with crystalline norms, showing that near-minimizers are close to a polyhedral set and that minimizers with small mass preserve crystalline structure.

Contribution

It proves a new stability inequality for crystalline Wulff energies and demonstrates that small mass minimizers retain crystalline polyhedral shapes, extending previous 2D results to higher dimensions.

Findings

All epsilon-minimizers are close to the convex polyhedron K.

The Wulff energy grows linearly away from the set of close polyhedra.

Small mass minimizers preserve the crystalline structure of the surface tension.

Abstract

Let $K$ be a convex polyhedron and $\mathscr F$ its Wulff energy, and let $\mathscr C(K)$ denote the set of convex polyhedra close to $K$ whose faces are parallel to those of $K$. We show that, for sufficiently small $\epsilon$, all $\epsilon$-minimizers belong to $\mathscr C(K)$. As a consequence of this result we obtain the following sharp stability inequality for crystalline norms: There exist $\gamma=\gamma(K,n)>0$ and $\sigma=\sigma(K,n)>0$ such that, whenever $|E|=|K|$ and $|E\Delta K|\le \sigma$ then $$ \mathscr F(E) - \mathscr F(K^a)\ge \gamma |E \Delta K^a| \qquad \text{for some }K^a \in \mathscr C(K). $$ In other words, the Wulff energy $\mathscr F$ grows very fast (with power $1$) away from the set $\mathscr C(K).$ The set $K^a \in \mathscr C(K)$ appearing in the formula above can be informally thought as a sort of "projection" of $E$ on the set $\mathscr C(K).$ Another corollary of our result is a very strong rigidity result for crystals: For crystalline surface tensions, minimizers of $\mathscr F(E)+\int_E g$ with small mass are polyhedra with sides parallel to the ones of $K$. In other words, for small mass, the potential energy cannot destroy the crystalline structure of minimizers. This extends to arbitrary dimensions a two-dimensional result obtained in [9].