Regularity conjecture for minimizers of the electrostatic Born–Infeld energy

Let $N$ be the ambient dimension, let $q>N$, let $\rho\in L^q_{loc}(\mathbb{R}^N)\cap\mathcal{X}^*$, and let $u_\rho$ be the minimizer associated with $\rho$. Regularity conjecture. The minimizer $u_\rho$ is $C^{1,\alpha}_{loc}(\mathbb{R}^N)$ for some $\alpha\in(0,1)$. This conjecture is motivated by the analogous regularity for the Poisson equation and the $p$-Laplacian, as well as by the radial example discussed in the paper; its general validity is not established in the supplied context.