Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model

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 $\partial \Omega$. The graph function $u_\rho : \Omega \rightarrow \mathbb{R}$ also represents the electric potential generated by a charge $\rho$ in electrostatic Born-Infeld theory. While $u_\rho$ minimizes the action $$ I_\rho(\psi) = \int_{\Omega} \Big( 1 - \sqrt{1-|D\psi|^2} \Big) \mathrm{d} x - \langle \rho, \psi \rangle $$ among competitors with $|D\psi| \le 1$, because of a lack of smoothness of the Lagrangian density when $|D\psi| = 1$ a direct approach via minimization may not produce a solution to the Euler-Lagrange equation (BI). In this paper, we study existence and regularity of $u_\rho$ for general $\rho$, in a bounded domain and in the entire $\mathbb{R}^m$. In particular, we find sufficient conditions to guarantee that $u_\rho$ solves (BI) and enjoys log-improved $W^{2,2}_{\mathrm{loc}}$ estimates, and we construct examples helping to identify sharp thresholds for the regularity of $\rho$ to ensure the validity of (BI). One of the main difficulties is the possible presence of light segments in the graph of $u_\rho$, which will be discussed in detail.