Spacelike graphs with prescribed mean curvature on exterior domains in the Minkowski spacetime

We consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a function defined on a domain equal to the complement in $R^n$ of the union of a finite number of bounded Lipschitz domains. The mean curvature $H=H(x,t)$ is assumed to have absolute value controlled from above by a locally bounded $L^p$-function, $pin[1,2n/(n+2)], $ngeq 3$.