Strong stability for the Wulff inequality with a crystalline norm

Abstract

Let $K$ be a convex polyhedron and $\mathcal{F}$ its Wulff energy, and let $\mathcal{C}(K)$ 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 $\mathcal{C}(K)$. 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 $$\mathcal{F}(E)-\mathcal{F}(K^a)\ge \gamma |E\Delta K^a|\text{for some }{K}^a\in \mathcal{C}(K).$$ In other words, the Wulff energy $\mathcal{F}$ grows very fast (with power $1$) away from the set $\mathcal{C}(K).$ The set $K^a\in \mathcal{C}(K)$ appearing in the formula above can be informally thought as a sort of "projection" of $E$ on the set $\mathcal{C}(K).$ Another corollary of our result is a very strong rigidity result for crystals: For crystalline surface tensions, minimizers of $\mathcal{F}(E)+{\int }_{E}g$ 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].

21 pages, 4 figures