  We extend the interior gradient estimate due to N. Korevaar and L. Simon for
solutions of the mean curvature equation from the case of Euclidean graphs to
the general case of Killing graphs. Our main application is the proof of
existence of Killing graphs with prescribed mean curvature function for
continuous boundary data, thus extending a result due to Dajczer, Hinojosa and
Lira. In addition, we prove the existence and uniqueness of radial graphs in
hyperbolic space with prescribed mean curvature function and asymptotic
boundary data at infinity.
