On the Quasi-Linear Elliptic PDE $-\mathrm{\nabla }\cdot (\mathrm{\nabla }u\mathrm{/}\sqrt{1-\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2})=4\pi {\sum }_k{a}_k{\delta }_{s_k}$ in Physics and Geometry

It is shown that for each finite number of Dirac measures supported at points $s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, there exists a unique real-valued Lipschitz function $u$, vanishing at infinity, which distributionally solves the quasi-linear elliptic partial differential equation of divergence form $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2})=4\pi\sum_{n=1}^N a_n \delta_{s_n}$\mathrm{.}Moreover,$u$isrealanalyticawayfromthe$s_n$. The result can be interpreted in at least two ways: (a) for any number of point charges of arbitrary magnitude and sign at prescribed locations $s_n$ in three-dimensional Euclidean space there exists a unique electrostatic field which satisfies the Maxwell-Born-Infeld field equations smoothly away from the point charges and vanishes as $\mathrm{\mid }s\mathrm{\mid }\to \mathrm{\infty }$; (b) for any number of integral mean curvatures assigned to locations $s_n$ there exists a unique asymptotically flat, almost everywhere space-like maximal slice with point defects of Minkowski spacetime, having lightcone singularities over the $s_n$ but being smooth otherwise, and whose height function vanishes as $\mathrm{\mid }s\mathrm{\mid }\to \mathrm{\infty }$. No struts between the point singularities ever occur.