Portal de Periódicos da CAPES

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

1997; Volume: 1; Issue: 3 Linguagem: Inglês

10.4310/ajm.1997.v1.n3.a6

1945-0036

Stanley Osher, Barry Merriman,

Point processes and geometric inequalities

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.