Existence of geodesics in static Lorentzian manifolds with convex boundary

We study some global geometric properties of a static Lorentzian manifold Lambda embedded in a differentiable manifold M, with possibly non-smooth boundary partial derivative Lambda. We prove a variational principle for geodesics in static manifolds, and using this principle we establish the existence of geodesics that do not touch partial derivative Lambda and that join two fixed points of Lambda. The results are obtained under a suitable completeness assumption for Lambda that generalizes the property of global hyperbolicity, and a weak convexity assumption on partial derivative Lambda. Moreover, under a non-triviality assumption on the topology of Lambda, we also get a multiplicity result for geodesics in Lambda joining two fixed points.