Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

This paper treats degenerate parabolic equations of second order uₜ + F (u, ² u) = 0 (14. 1) related to differential geometry, where ∇ stands for spatial derivatives of u = ut, x) in x ∈ R n, and u t represents the partial derivative of u in time t. We are especially interested in the case when (1. 1) is regarded as an evolution equation for level surfaces of u. It turns out that (1. 1) has such a property if F has a scaling invariance F (p, X + p p) = F (p. X), \, \, \, \, \, \, > 0, \, \, R (14. 2) for a nonzero p ∈ R n and a real symmetric matrix X, where ⊗ denotes a tensor product of vectors in R n. We say (1. 1) is geometric if F satisfies (1. 2). A typical example is uₜ - | { u |div (u/| u |) = 0, (14. 3) where ∇u is the (spatial) gradiant of u. Here ∇u/|∇u| is a unit normal to a level surface of u, so div (∇u/|∇u|) is its mean curvature unless ∇u vanishes on the surface. Since u t /\∇u is a normal velocity of the level surface, (1. 3) implies that a level surface of solution u of (1. 3) moves by its mean curvature unless ∇u vanishes on the surface. We thus call (1. 3) the mean curvature flow equation in this paper.