In this talk we present some recent results concerning the regularity of the unique weak solution vanishing at infinity of the prescribed mean curvature equation in the Lorentz-Minkowski space for spacelike hypersurfaces, when the mean curvature belongs to $L^p(R^N)$, with $p>N$. This equation is also known as the ``Born-Infeld'' equation, as it comes from the nonlinear model of electromagnetism introduced by M. Born and L. Infeld, but it also plays a crucial role in Relativity. In the first part of the talk we will show a new gradient estimate for smooth solutions of the prescribed mean curvature equation and prove that, under our assumptions, the unique minimizer of the Born-Infeld energy, which is a priori only Lipschitz continuous, is actually a strictly spacelike weak solution of class $W^{2,p}$. In the second part the we will discuss some other related results concerning the existence of spacelike radial graphs of prescribed mean curvature and some open problems. These results are collected in a series of joint works with Prof. D. Bonheure (Université Libre de Bruxelles).