Let (M,g) be a Riemannian manifold and (\({\mathbb{R}}\times M, -\alpha^ 2dt^ 2\oplus g)\) a static Lorentzian warped product in which M is closed. The author proves that the Dirichlet problem for the mean curvature operator H has a (unique up to a constant) spacelike \(C^{2,\delta}\) solution if and only if \(\int_{M}H\alpha dv=0.\)