Abstract
It is shown that for each finite number N of Dirac measures \({\delta_{s_n}}\) supported at points \({s_n \in {\mathbb R}^3}\) with given amplitudes \({a_n \in {\mathbb R} \backslash\{0\}}\) there exists a unique real-valued function \({u \in C^{0, 1}({\mathbb R}^3)}\), 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, \({u \in C^{\omega}({\mathbb R}^3\backslash \{s_n\}_{n=1}^N)}\). The result can be interpreted in at least two ways: (a) for any number N 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| → ∞; (b) for any number N of integral mean curvatures assigned to locations \({s_n \in {\mathbb R}^3 \subset{\mathbb R}^{1, 3}}\) there exists a unique asymptotically flat, almost everywhere space-like maximal slice with point defects of Minkowski spacetime \({{\mathbb R}^{1, 3}}\), having lightcone singularities over the s n but being smooth otherwise, and whose height function vanishes as |s| → ∞. No struts between the point singularities ever occur.
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Communicated by P. T. Chruściel
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Kiessling, M.KH. 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. Commun. Math. Phys. 314, 509–523 (2012). https://doi.org/10.1007/s00220-012-1502-3
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DOI: https://doi.org/10.1007/s00220-012-1502-3