Entire spherically symmetric spacelike graphs with prescribed mean curvature function in Schwarzschild and Reissner-Nordstrom spacetimes

For each spacetime of a family of static spacetimes, we prove the existence of entire spherically symmetric spacelike graphs with prescribed mean curvature function. In particular, classical Schwarzschild and Reissner-Nordstrom spacetimes are considered. In both cases, the entire spacelike graph asymptotically approaches the event horizon. Spacelike graphs of constant mean curvature remain as a particular situation in the existence results, obtaining explicit expressions for the solutions. The proof of the results is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball, together with the obtention of a suitable bound for the length of the gradient of a solution which permits the prolongability to the whole space.