Existence and Regularity for Prescribed Lorentzian Mean Curvature Hypersurfaces, and the Born–Infeld Model

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 \(\partial \Omega \), which solve

The graph function also represents the electric potential generated by a charge \(\rho \) in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer \(u_\rho \) of the associated action

$$\begin{aligned} I_\rho (\psi ) \doteq \int _{\Omega } \Big ( 1 - \sqrt{1-|D\psi |^2} \Big ) \textrm{d}x - \langle \rho , \psi \rangle \end{aligned}$$

among functions \(\psi \) satisfying \(|D\psi | \le 1\), by the lack of smoothness of the Lagrangian density for \(|D\psi | = 1\) one cannot guarantee that \(u_\rho \) satisfies the Euler-Lagrange equation (\(\mathcal{B}\mathcal{I}\)). A chief difficulty comes from the possible presence of light segments in the graph of \(u_\rho \). In this paper, we investigate the existence of a solution for general \(\rho \). In particular, we give sufficient conditions to guarantee that \(u_\rho \) solves (\(\mathcal{B}\mathcal{I}\)) and enjoys \(\log \)-improved energy and \(W^{2,2}_\textrm{loc}\) estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of \(\rho \) to ensure the solvability of (\(\mathcal{B}\mathcal{I}\)).