Uniqueness without growth condition for the
mean curvature equation with radial initial data

Samuel Biton, Emmanuel Chasseigne
and Olivier Ley

We prove the uniqueness of a
solution to the mean curvature equation for graphs
for any radial continuous initial data.
The existence of a smooth solution to this problem comes
from the work of Ecker and Huisken.
We obtain in addition that the solution is radial.
We point out that existence and uniqueness hold without
any growth restriction on the solution
or the initial data, a situation which is rather different to the
related stationary problem: in this case, we show there is a
limiting growth above which uniqueness does not hold
anymore. An application of the uniqueness result to the
evolution by mean curvature of entire radial graphs
is given.