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.