# Uniqueness and Existence of Viscosity Solutions of Generalized mean Curvature Flow Equations

• Yun-Gang Chen
• Yoshikazu Giga
• Shun’Ichi Goto
Chapter

## 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 xR 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 pR 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.

## Keywords

Parabolic Equation Viscosity Solution Level Surface Comparison Theorem Evolution Family
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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