Equilibrium Shapes for Planar Crystals in an External Field

Abstract

The equilibrium shape of a two-dimensional crystal in a convex background potential g(x) is analyzed. For g = 0 the shape of minimum energy may be deduced from surface tension via the Wulff construction, but if g is not constant, little is known beyond the case of a crystal sitting in a uniform field. Only an unpublished result of Okikiolu shows each connected component of the equilibrium crystal to be convex. Here it will be shown that any such component minimizes energy uniquely among convex sets of its area. If the Wulff shape and g(x) are symmetric under x ↔ −x, it follows that the equilibrium crystal is unique, convex and connected. This last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature-driven flow. Subsequent work with Felix Otto shows – without assuming symmetry –that no equilibrium crystal has more than two convex components.