The Wulff shape as the asymptotic limit of a growing crystalline interface

We present a proof of a conjecture made in the field of crystal growth. Namely, for an initial state consisting of any number of growing crystals moving outwards with normal velocity given to be 7 (n), for ft the unit outwards normal, then the asymptotic growth shape is a Wulff crystal, appropriately scaled in time. This shape minimizes the surface energy, which is the surface integral of j (n), for a given volume. The proof works in any number of dimensions. Additionally, we develop a new approach for obtaining the Wulff shape by minimizing the surface energy divided by the enclosed volume to the ^ power in R d. We show that if we evolve a convex surface (not enclosing a Wulff shape) under the motion described above, that the quantity to be minimized strictly decreases to its minimum as time increases. We have thus discovered a link between this surface evolution and this (generally nonconvex) energy minimization. A generalized Huyghen's principle is obtained. Finally, given the asymptotic shape we also obtain the associated (unique) convex 7 (n). The key technical tool is the level set method and the theory and characterization of viscosity solutions to Hamilton-Jacobi equations.