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In this paper, we consider the electrostatic Born-Infeld model\begin{equation*} \tag{$\mathcal{BI}$} \left\{ \begin{array}{rcll}-\operatorname{div}\left(\displaystyle\frac{\nabla \phi}{\sqrt{1-|\nabla\phi|^2}}\right)&=& \rho & \hbox{in }\mathbb{R}^N, \\[6mm]\displaystyle\lim_{|x|\to \infty}\phi(x)&=& 0 \end{array} \right.\end{equation*} where $\rho$ is a charge distribution on the boundary of abounded domain $\Omega\subset \mathbb{R}^N$. We are interested in itsequilibrium measures, i.e. charge distributions which minimize theelectrostatic energy of the corresponding potential among all possibledistributions with fixed total charge. We prove existence of equilibriummeasures and we show that the corresponding equilibrium potential is unique andconstant in $\overline \Omega$. Furthermore, for smooth domains, we obtain theuniqueness of the equilibrium measure, we give its precise expression, and weverify that the equilibrium potential solves ($\mathcal{BI}$). Finally wecharacterize balls in $\mathbb{R}^N$ as the unique sets among all bounded$C^{2,\alpha}$-domains $\Omega$ for which the equilibrium distribution is aconstant multiple of the surface measure on $\partial\Omega$. The same resultsare obtained also for Taylor approximations of the electrostatic energy.

Equilibrium measures and equilibrium potentials in the Born-Infeld model