Motion of a Set by the Curvature of Its Boundary

Abstract

The study of a crystal shrinking or growing in a melt gives rise to equations relating the normal velocity of the motion to the curvature of the crystal boundary. Often these equations are anisotropic, indicating the preferred directions of the crystal structure. In the isotropic case this equation is called the curve shortening or the mean curvature flow equation, and has been studied by differential geometric tools. In general, there are no classical solutions to these equations. In this paper we develop a weak theory for the generalized mean curvature equation using the newly developed theory of viscosity solutions. Our approach is closely related to that of Osher and Sethian, Chen, Giga and Goto, and Evans and Spruck, who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation. Although we use their results in an essential way, we give an intrinsic definition. Our main results are the existence of a solution, large time asymptotics of this solution, and its connection to the level set solution of Osher and Sethian, Chen, Giga and Goto, and Evans and Spruck. In general there is no uniqueness, even for classical solutions, but we prove a uniqueness result under restrictive assumptions. We also construct a class of explicit solutions which are dilations of Wulff crystals.