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

Abstract

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 $-\nabla \cdot (\nabla u/\sqrt{1-|\nabla u{|}^2})=4\pi {\sum }_{n=1}^N{a}_n{\delta }_{s_n}$. Moreover, $u$ is real analytic away from the $s_n$. 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 $|s|\to \infty$; (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 $|s|\to \infty$. No struts between the point singularities ever occur.

This is the preprint of the version published in 2012 in Commun. Math. Phys. PLUS an errata which has been accepted 08/13/2018 for publication in Commun. Math. Phys

References in corpus (3)

• On the Static Spacetime of a Single Point Charge
• Born-Infeld electrostatics in the complex plane
• Convergent perturbative power series solution of the stationary Maxwell--Born--Infeld field equations with regular sources

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