  The present article proposes a rigorous derivation of the Boltzmann equation
in the half-space. We show an analog of the Lanford's theorem in this domain,
with specular reflection boundary condition, stating the convergence in the low
density limit of the first marginal of the density function of a system of N
hard spheres towards the solution of the Boltzmann equation associated to the
initial data corresponding to the initial state of the one-particle-density
function. The original contributions of this work consist in two main points:
the rigorous definition of the collision operator and of the functional space
in which the BBGKY hierarchy is solved in a strong sense; and the adaptation to
the case of the half-space of the control of the recollisions performed by
Gallagher, Saint-Raymond and Texier, which is a crucial step to obtain the
Lanford's theorem.
