On the electrostatic Born-Infeld equation with extended charges

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to the electrostatic Born-Infeld equation with extended charge densities, using variational methods and analyzing special cases like point charges and coupled models.

Contribution

It provides a comprehensive variational framework for solving the Born-Infeld equation with general charge densities, including symmetry and regularity results, and discusses extensions to coupled nonlinear models.

Findings

Existence and uniqueness of solutions for broad charge densities.

Regularity results for smooth and locally bounded charges.

Analysis of point charge superpositions and related models.

Abstract

In this paper, we deal with the electrostatic Born-Infeld equation \begin{equation}\label{eq:BI-abs} \tag{$\mathcal{BI}$} \left\{ \begin{array}{ll} -\operatorname{div}\left(\displaystyle\frac{\nabla \phi}{\sqrt{1-|\nabla \phi|^2}}\right)= \rho, & \hbox{in } \mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty}\phi(x)= 0, \end{array} \right. \end{equation} where $\rho$ is an assigned extended charge density. We are interested in the existence and uniqueness of the potential $\phi$ and finiteness of the energy of the electrostatic field $-\nabla \phi$. We first relax the problem and treat it with the direct method of the Calculus of Variations for a broad class of charge densities. Assuming $\rho$ is radially distributed, we recover the weak formulation of \eqref{eq:BI-abs} and the regularity of the solution of the Poisson equation (under the same smootheness assumptions). In the case of a locally bounded charge, we also recover the weak formulation without assuming any symmetry. The solution is even classical if $\rho$ is smooth. Then we analyze the case where the density $\rho$ is a superposition of point charges and discuss the results in [Kiessling, Comm. Math. Phys. 314 (2012), 509--523]. Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein-Gordon equation with the Born-Infeld theory.