Published Paper

Inserted: 2 apr 2020
Last Updated: 21 aug 2024

Journal: Comm. Pure Appl. Math.
Year: 2022

Abstract:

Let \( K \) be a convex polyhedron and \( F \) its Wulff energy, and let \( C(K) \) denote the set of convex polyhedra close to \( K \) whose faces are parallel to those of \( K \). We show that, for sufficiently small \( \varepsilon \), all \( \varepsilon \)-minimizers belong to \( C(K) \).

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 \(
E \Delta K
\leq \sigma \), then \[ F(E) - F(K_a) \geq \gamma
E \Delta K_a
\] for some \( K_a \in C(K) \).

In other words, the Wulff energy \( F \) grows very fast (with power 1) away from \( C(K) \). The set \( K_a \in C(K) \) appearing in the formula above can be informally thought of as a sort of “projection” of \( E \) onto \( C(K) \).

Another corollary of our result is a very strong rigidity result for crystals: For crystalline surface tensions, minimizers of \( F(E) + \int_E g \) with small mass are polyhedra with sides parallel to those of \( K \). In other words, for small mass, the potential energy cannot destroy the crystalline structure of minimizers.

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• Strong-stability-for-the-wulff-inequality-with-a-crystalline-norm.pdf