Calculus of Variations and Geometric Measure Theory

G. Bellettini - G. Fusco

Some Aspects of the Dynamic of $V=H-\bar H$

created on 11 Nov 1996
modified on 08 Feb 2001


Inserted: 11 nov 1996
Last Updated: 8 feb 2001

Year: 1996


We consider the evolution of a surface $\Gamma(t)$ according to the equation $V = H-\bar H$, where $V$ is the normal velocity of $\Gamma(t)$, $H$ is the sum of the two principal curvatures and $\bar H$ is the average of $H$ on $\Gamma(t)$. We study the case where $\Gamma(t)$ intersects orthogonally a fixed surface $\Sigma$, and discuss some aspects of the dynamics of $\Gamma$ under the assumption that the volume of the region enclosed between $\Gamma(t)$ and $\Sigma$ is small. We show that, in this case, if $\Gamma(0)$ is near a hemisphere, $\Gamma(t)$ keeps its almost hemispherical shape and slides on $\Sigma$ crawling approximately along orbits of the tangential gradient $\nabla H_\Sigma$ of the sum $H_\Sigma$ of the two principal curvatures of $\Sigma$. We also show that, if $\bar p \in S$ is a nondegenerate zero of the gradient of $\nabla H_\Sigma$ and $a>0$ is sufficiently small, then there is a surface of constant mean curvature which is near a hemisphere of radius a with center near $\bar p$ and intersects $\Sigma$ orthogonally.