Evolution of Nonparametric Surfaces with Speed Depending on Curvature, III. Some Remarks on Mean Curvature and Anisotropic flows

Abstract

This paper is a sequel to our paper [OU] where we investigated questions concerning solvability and asymptotic behavior of solutions to the mean curvature evolution problem

$$ {u_t} = \sqrt {1 + {{\left| {Du} \right|}^2}} H\left( u \right)\;in\quad \Omega \times \left( {0,\infty } \right),$$

(1.1)

$$ u\left( {x,t} \right) = 0\quad on\quad \partial \Omega \times \left[ {0,\infty } \right),$$

(1.2)

$$ u\left( {x,0} \right) = {u_0}\left( x \right)\quad in\quad \overline {\Omega ,} \quad {u_0} \in C_0^\infty \left( {\overline \Omega } \right)$$

(1.3)

where Ω is a bounded domain in R ⁿ, n ≥ 2, with C^∞ boundary ∂Ω, H is the mean curvature operator.