On the regularity of the minimizer of the electrostatic Born-Infeld energy

Abstract

We consider the electrostatic Born-Infeld energy \begin{equation*} \int_{\mathbb{R}^N}\left(1-{\sqrt{1-|\nabla u|^2}}\right)\, dx -\int_{\mathbb{R}^N}ρu\, dx, \end{equation*} where $\rho \in L^m({\mathbb{R}}^N)$ is an assigned charge density, $m\in [1,2_{\ast }]$, $2_{\ast }:=\frac{2N}{N+2}$, $N\ge 3$. We prove that if $\rho \in L^q({\mathbb{R}}^N)$ for $q>2N$, the unique minimizer $u_{\rho }$ is of class $W_{loc}^{2,2}({\mathbb{R}}^N)$. Moreover, if the norm of $\rho$ is sufficiently small, the minimizer is a weak solution of the associated PDE \begin{equation}\label{eq:BI-abs} \tag{$\mathcal{B}\mathcal{I}$} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= ρ\quad\hbox{in }\mathbb{R}^N, \end{equation} with the boundary condition ${\lim }_{|x|\to \infty }u(x)=0$ and it is of class $C_{loc}^{1,\alpha }({\mathbb{R}}^N)$, for some $\alpha \in (0,1)$.