On the Born-Infeld equation for electrostatic fields with a superposition of point charges

Abstract

In this paper, we study the static Born-Infeld equation $$ -\mathrm{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)=\sum_{k=1}^n a_kδ_{x_k}\quad\mbox{in }\mathbb R^N,\qquad \lim_{|x|\to\infty}u(x)=0, $$ where $N\ge 3$, $a_k\in \mathbb{R}$ for all $k=1,\dots ,n$, $x_k\in {\mathbb{R}}^N$ are the positions of the point charges, possibly non symmetrically distributed, and ${\delta }_{x_k}$ is the Dirac delta distribution centered at $x_k$. For this problem, we give explicit quantitative sufficient conditions on $a_k$ and $x_k$ to guarantee that the minimizer of the energy functional associated to the problem solves the associated Euler-Lagrange equation. Furthermore, we provide a more rigorous proof of some previous results on the nature of the singularities of the minimizer at the points $x_k$'s depending on the sign of charges $a_k$'s. For every $m\in \mathbb{N}$, we also consider the approximated problem $$ -\sum_{h=1}^mα_hΔ_{2h}u=\sum_{k=1}^n a_kδ_{x_k}\quad\mbox{in }\mathbb R^N, \qquad\lim_{|x|\to\infty}u(x)=0 $$ where the differential operator is replaced by its Taylor expansion of order $2m$, see (2.1). It is known that each of these problems has a unique solution. We study the regularity of the approximating solution, the nature of its singularities, and the asymptotic behavior of the solution and of its gradient near the singularities.

24 pages

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