The electrostatic Born-Infeld equations with integrable charge densities

Abstract

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}ρv\ 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)=ρ,\end{equation*} provided that $\rho$ is in the dual space of the solution space and $\rho \in L^p({\mathbb{R}}^n)$, for some $p>n\ge 3$. Moreover, we have $u\in C^{1,\alpha }({\mathbb{R}}^n)$ for some $\alpha \in (0,1)$.

arXiv comment21 pages, clarified notation, typos corrected, reorganized proof of Theorem 4.2