Solvability and regularity for the electrostatic Born-Infeld equation with general charges

In electrostatic Born-Infeld theory, the electric potential u ρ generated by a charge distribution ρ in R m (typically, a Radon measure) minimizes the action ∫ R m ( 1 − √ 1 − ∣ D ψ ∣ 2 ) d x − ⟨ ρ , ψ ⟩ among functions which decay at infinity and satisfy ∣ D ψ ∣≤ 1 . Formally, its Euler-Lagrange equation prescribes ρ as being the Lorentzian mean curvature of the graph of u ρ in Minkowski spacetime L m + 1 . However, because of the lack of regularity of the functional when ∣ D ψ ∣= 1 , whether or not u ρ solves the Euler-Lagrange equation and how regular is u ρ are subtle issues that were investigated only for few classes of ρ . In this paper, we study both problems for general sources ρ , in a bounded domain with a Dirichlet boundary condition and in the entire R m . In particular, we give sufficient conditions to guarantee that u ρ solves Ethe uler-Lagrange equation and enjoys improved W 2 , 2 loc estimates, and we construct examples helping to identify sharp thresholds for the regularity of ρ to ensure the validity of the Euler-Lagrange equation. One of the main difficulties is the possible presence of light segments in the graph of u ρ , which will be discussed in detail.