A sharp gradient estimate and $W^{2,q}$ regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely \begin{equation*} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \rho \quad \hbox{in }\mathbb{R}^N, \end{equation*} where $N\geq 3$. We first prove a new gradient estimate for classical solutions with smooth data $\rho$. As a consequence we obtain that the unique weak solution of the equation satisfying a homogeneous boundary condition at infinity is locally of class $W^{2,q}$ and strictly spacelike in $\mathbb{R}^N$, provided that $\rho\in L^q(\mathbb{R}^N) \cap L^m(\mathbb{R}^N)$ with $q>N$ and $m\in[1,\frac{2N}{N+2}]$.