Abstract. We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergenceform uniformly elliptic systems. The estimates are optimal in dimensions larger than three and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition.

The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier-Stokes equations in the half-space R 3 + . Such solutions are sometimes called Lemarié-Rieusset solutions in the whole space R 3 . The main tool in our work is an explicit representation formula for the pressure, which is decomposed into a Helmholtz-Leray part and a harmonic part due to the boundary. We also explain how our result enables to reprove the blow-up of the scale-critical L 3 (R 3 + ) norm obtained by Barker and Seregin for solutions developing a singularity in finite time.