On the Electrostatic Born–Infeld Equation with Extended Charges

Abstract

In this paper, we deal with the electrostatic Born–Infeld equation

$$\left\{\begin{array}{ll}-\operatorname{div} \left(\displaystyle\frac{\nabla\phi}{\sqrt{1-|\nabla \phi|^2}} \right)= \rho \quad{in} \mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty} \phi(x)= 0,\end{array}\right. \quad \quad \quad \quad ({\mathcal{BI}})$$

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 ${({\mathcal{BI}})}$ and the regularity of the solution of the Poisson equation (under the same smoothness 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, Commun Math Phys 314:509–523, 2012). Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein–Gordon equation with the Born–Infeld theory.