  We study statistical properties of a continuum model of polynuclear surface
growth on an infinite substrate. We develop a self-consistent mean-field theory
which is solved to deduce the growth velocity and the extremal behavior of the
coverage. Numerical simulations show that this theory gives an improved
approximation for the coverage compare to the previous linear recursion
relations approach. Furthermore, these two approximations provide useful upper
and lower bounds for a number of characteristics including the coverage, growth
velocity, and the roughness exponent.
