  The Branching Brownian Motions (BBM) are particles performing independent
Brownian motions in $\mathbb R$ and each particle at rate 1 creates a new
particle at her current position; the newborn particle increments and
branchings are independent of the other particles. The $N$-BBM starts with $N$
particles and at each branching time, the leftmost particle is removed so that
the total number of particles is $N$ for all times. The $N$-BBM was proposed by
Maillard and belongs to a family of processes introduced by Brunet and Derrida.
We fix a density $\rho$ with a left boundary $L=\sup\{r\in\mathbb R:
\int_r^\infty \rho(x)dx=1\}>-\infty$ and let the initial particle positions be
iid continuous random variables with density $\rho$. We show that the empirical
measure associated to the particle positions at a fixed time $t$ converges to
an absolutely continuous measure with density $\psi(\cdot,t)$, as $N\to\infty$.
The limit $\psi$ is solution of a free boundary problem (FBP) when this
solution exists. The existence of solutions for finite time-intervals has been
recently proved by Lee.
