In this paper, we consider the electrostatic Born-Infeld model($\mathcal{BI}$)$\{\begin{array}{cccc}\hfill -\mathrm{div}\left(\frac{\mathrm{\nabla }\varphi }{\sqrt{1-|\mathrm{\nabla }\varphi {|}^2}}\right)& \hfill =\hfill & \rho \hfill & \text{in }{\mathbb{R}}^N,\hfill \\ \hfill \underset{|x|\to \infty }{\lim }\varphi (x)& \hfill =\hfill & 0\hfill \end{array}$ where ρ is a charge distribution on the boundary of a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, with $N⩾3$. We are interested in its equilibrium measures, i.e. charge distributions which minimize the electrostatic energy of the corresponding potential among all possible distributions with fixed total charge. We prove existence of equilibrium measures and we show that the corresponding equilibrium potential is unique and constant in $\bar{\mathrm{\Omega }}$. Furthermore, for smooth domains, we obtain the uniqueness of the equilibrium measure, we give its precise expression, and we verify that the equilibrium potential solves ($\mathcal{BI}$). Finally we characterize balls in ${\mathbb{R}}^N$ as the unique sets among all bounded $C^{2,\alpha }$-domains Ω for which the equilibrium distribution is a constant multiple of the surface measure on ∂Ω. The same results are obtained also for Taylor approximations of the electrostatic energy.

在本文中，我们考虑静电Born-Infeld模型（$\mathcal{双}$）$\{\begin{array}{cccc}\hfill -\mathrm{div}（\frac{\mathrm{\nabla }\varphi }{\sqrt{\mathrm{1个}-|\mathrm{\nabla }\varphi {|}^2}}）& \hfill =\hfill & \rho \hfill & \text{在 }{\mathbb{[R}}^{ñ}，\hfill \\ \hfill \underset{|X|\to \infty }{\mathrm{林}}\varphi （X）& \hfill =\hfill & 0\hfill \end{array}$其中ρ是有界域边界上的电荷分布$\mathrm{\Omega }\subset {\mathbb{[R}}^{ñ}$，带有 $ñ⩾3$。我们对它的平衡度量感兴趣，即在总电荷固定的所有可能分布中，使相应电位的静电能最小化的电荷分布。我们证明了平衡测度的存在，并且证明了相应的平衡势在$\stackrel{〜}{\mathrm{\Omega }}$。此外，对于平滑域，我们获得了平衡测度的唯一性，给出了它的精确表达式，并验证了平衡势能满足（$\mathcal{双}$）。最后，我们在${\mathbb{[R}}^{ñ}$ 作为所有边界中的唯一集合 $C^{2，\alpha }$-区域Ω，其平衡分布是measureΩ上表面度量的恒定倍数。对于静电能的泰勒近似也获得相同的结果。