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

Abstract

This paper treats degenerate parabolic equations of second order $$u_t + F(\nabla u,\nabla ^2 u) = 0$$ (14.1) related to differential geometry, where ∇ stands for spatial derivatives of u = u{t,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 $$F(\lambda p,\lambda X + \sigma p \otimes p) = \lambda F(p.X),\,\,\,\,\,\,\lambda > 0,\,\,\sigma \in \mathbb{R}$$ (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 $$u_t - \left| {\nabla u} \right|div(\nabla u/\left| {\nabla u} \right|) = 0,$$ (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.