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
