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The present article proposes a rigorous derivation of the Boltzmann equationin 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 lowdensity limit of the first marginal of the density function of a system of Nhard spheres towards the solution of the Boltzmann equation associated to theinitial data corresponding to the initial state of the one-particle-densityfunction. The original contributions of this work consist in two main points:the rigorous definition of the collision operator and of the functional spacein which the BBGKY hierarchy is solved in a strong sense; and the adaptation tothe case of the half-space of the control of the recollisions performed byGallagher, Saint-Raymond and Texier, which is a crucial step to obtain theLanford's theorem.

About Lanford's theorem in the half-space with specular reflection