On the Quasi-Linear Elliptic PDE $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2}) = 4π\sum_k a_k δ_{s_k}$ in Physics and Geometry

Details On the Quasi-Linear Elliptic PDE $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2}) = 4π\sum_k a_k δ_{s_k}$ in Physics and Geometry On the Quasi-Linear Elliptic PDE $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2}) = 4π\sum_k a_k δ_{s_k}$ in Physics and Geometry

2011-07-20

arxiv.org/abs/1107.4126v2

Physics

Mathematical Physics

18 pages; referee's comments incorporated in this revised version; accepted for publication in Commun. Math. Phys

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

On the Quasi-Linear Elliptic PDE $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2}) = 4π\sum_k a_k δ_{s_k}$ in Physics and Geometry does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the Quasi-Linear Elliptic PDE $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2}) = 4π\sum_k a_k δ_{s_k}$ in Physics and Geometry, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Quasi-Linear Elliptic PDE $-\nabla\cdot(\nabla{u}/\sqrt{1-|\nabla{u}|^2}) = 4π\sum_k a_k δ_{s_k}$ in Physics and Geometry will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-687190