A strong maximum principle for weak solutions of quasi-linear elliptic equations with applications to Lorentzian and Riemannian geometry

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and supersolutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for C0-space-like hypersurfaces in a Lorentzian manifold. As one application, a Lorentzian warped product splitting theorem is given.