We study the minimizer of the electrostatic Born--Infeld energy \begin{ equation*} \int_{ \mathbb{ R}^n}1-\sqrt{ 1-|D v|^2}\ dx-\int_{ \mathbb{ R}^n}\rho v\ dx, \end{ equation*} which vanishes at infinity. We show that the minimizer $u$ is strictly spacelike and it is a weak solution to \begin{ equation*}-\operatorname{ div}\Big(\frac{ D u}{ \sqrt{ 1-|D u|^2}}\Big)=\rho,\end{ equation*} provided that $\rho$ is in the dual space of the solution space and $\rho\in L^p(\mathbb{ R}^n)$, for some $p>n\geq 3$. Moreover, we have $u\in C^{ 1,\alpha}(\mathbb{ R}^n)$ for some $\alpha\in (0,1)$.
Categories: math.AP math-ph math.MP