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 $u_{\rho }:\Omega \to \mathbb{R}$ also represents the electric potential generated by a charge $\rho$ in electrostatic Born-Infeld theory. While $u_{\rho }$ minimizes the action $$I_{\rho }(\psi )={\int }_{\Omega }(1-\sqrt{1-|D\psi {|}^2})\mathrm{d}x-⟨\rho ,\psi ⟩$$ 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 $W_{\mathrm{loc}}^{2,2}$ 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.

72 pages. Final version, to appear in Annals of PDE. More explanations added (especially in the Introduction and in Subsection 4.4). Package axessibility included to make the paper available to visually impaired people

References in corpus (3)

• On the electrostatic Born-Infeld equation with extended charges
• On the regularity of the minimizer of the electrostatic Born-Infeld energy
• On the Born-Infeld equation for electrostatic fields with a superposition of point charges

Cited by in corpus (1)

• Compactness via monotonicity in nonsmooth critical point theory, with application to Born-Infeld type equations