  Given a measure $\rho$ on a domain $\Omega \subset \mathbb{R}^m$, we study
spacelike graphs over $\Omega$ in Minkowski space with Lorentzian mean
curvature $\rho$ and Dirichlet boundary condition on $\partial \Omega$. The
graph function $u_\rho : \Omega \rightarrow \mathbb{R}$ also represents the
electric potential generated by a charge $\rho$ in electrostatic Born-Infeld
theory. While $u_\rho$ minimizes the action
  $$
  I_\rho(\psi) = \int_{\Omega} \Big( 1 - \sqrt{1-|D\psi|^2} \Big) \mathrm{d} x
- \langle \rho, \psi \rangle
  $$ among competitors with $|D\psi| \le 1$, because of a lack of smoothness of
the Lagrangian density when $|D\psi| = 1$ a direct approach via minimization
may not produce a solution to the Euler-Lagrange equation (BI). In this paper,
we study existence and regularity of $u_\rho$ for general $\rho$, in a bounded
domain and in the entire $\mathbb{R}^m$. In particular, we find sufficient
conditions to guarantee that $u_\rho$ solves (BI) and enjoys log-improved
$W^{2,2}_{\mathrm{loc}}$ estimates, and we construct examples helping to
identify sharp thresholds for the regularity of $\rho$ to ensure the validity
of (BI). One of the main difficulties is the possible presence of light
segments in the graph of $u_\rho$, which will be discussed in detail.
