*Published Paper*

**Inserted:** 22 dec 2021

**Last Updated:** 17 aug 2024

**Journal:** Annals of PDE

**Volume:** 10

**Number:** 4

**Pages:** 86

**Year:** 2024

**Doi:** 10.1007/s40818-023-00167-4

**Abstract:**

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 $\partial \Omega$. These solve the (BI) equation \[ -{\rm div} \left( \frac{Du}{\sqrt{1- \vert Du \vert^2}} \right) = \rho \qquad \text{ in } \, \Omega, \] 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 \[ I_\rho(\psi) \doteq \int_{\Omega} \Big( 1 - \sqrt{1-\vert D\psi\vert^2} \Big) {\rm d} x - \langle \rho, \psi \rangle \] among functions $\psi$ satisfying $\vert D\psi \vert \le 1$, by the lack of smoothness of the Lagrangian density for $\vert D\psi\vert = 1$ one cannot guarantee that $u_\rho$ satisfies (BI). 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 (BI) and enjoys $\log$-improved energy and $W^{2,2}_{\rm loc}$ estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of $\rho$ to ensure the solvability of (BI).