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

Given a measure \(\rho \) $\rho$ on a domain \(\Omega \subset {\mathbb {R}}^m\) $\mathrm{\Omega }\subset {\mathbb{R}}^m$ , we study spacelike graphs over \(\Omega \) $\mathrm{\Omega }$ in Minkowski space with Lorentzian mean curvature \(\rho \) $\rho$ and Dirichlet boundary condition on \(\partial \Omega \) $\partial \mathrm{\Omega }$ , which solve The graph function also represents the electric potential generated by a charge \(\rho \) $\rho$ in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer \(u_\rho \) $u_{\rho }$ of the associated action \(\begin{aligned} I_\rho (\psi ) \doteq \int _{\Omega } \Big ( 1 - \sqrt{1-|D\psi |^2} \Big ) \textrm{d}x - \langle \rho , \psi \rangle \end{aligned}\) $$\begin{array}{c}\hfill I_{\rho }(\psi )\doteq {\int }_{\mathrm{\Omega }}(1-\sqrt{1-{|D\psi |}^2})\text{d}x-⟨\rho ,\psi ⟩\end{array}$$ among functions \(\psi \) $\psi$ satisfying \(|D\psi | \le 1\) $|D\psi |\le 1$ , by the lack of smoothness of the Lagrangian density for \(|D\psi | = 1\) $|D\psi |=1$ one cannot guarantee that \(u_\rho \) $u_{\rho }$ satisfies the Euler-Lagrange equation ( \(\mathcal{B}\mathcal{I}\) $\mathcal{B}\mathcal{I}$ ). A chief difficulty comes from the possible presence of light segments in the graph of \(u_\rho \) $u_{\rho }$ . In this paper, we investigate the existence of a solution for general \(\rho \) $\rho$ . In particular, we give sufficient conditions to guarantee that \(u_\rho \) $u_{\rho }$ solves ( \(\mathcal{B}\mathcal{I}\) $\mathcal{B}\mathcal{I}$ ) and enjoys \(\log \) $log$ -improved energy and \(W^{2,2}_\textrm{loc}\) $W_{\text{loc}}^{2,2}$ estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of \(\rho \) $\rho$ to ensure the solvability of ( \(\mathcal{B}\mathcal{I}\) $\mathcal{B}\mathcal{I}$ ).