Extremal curves on Riemannian and Lorentzian manifolds with boundary

Let M be a Riemannian manifold. Trajectories joining two points, closed geodesics and trajectories of particles under a potential (having fixed period or fixed energy) are critical points of functionals bounded from below on some Hilbert manifolds. By using some penalization techniques, it it possible to prove existence (and multiplicity in some cases) of such critical points, even when M has boundary (under some convexity assumptions). If M is a Lorentzian manifold, the correspondent functionals are strongly indefinite, nevertheless some variational principles allow one to extend previous result to stationary Lorentzian manifold with boundary such as Schwarzschild, Reissner-Nordstrom and Kerr spacetimes.