Preprint

Inserted: 22 dec 2021
Last Updated: 23 dec 2021

Pages: 71
Year: 2021

Abstract:

In electrostatic Born-Infeld theory, the electric potential $u_\rho$ generated by a charge distribution $\rho$ in $\mathbb{R}^m$ (typically, a Radon measure) minimizes the action \[ \int_{\mathbb{R}^m} \Big( 1 - \sqrt{1-\mid D\psi\mid^2} \Big) \mathrm{d} x - \langle \rho, \psi \rangle \] among functions which decay at infinity and satisfy $\mid D\psi\mid \le 1$. Formally, its Euler-Lagrange equation prescribes $\rho$ as being the Lorentzian mean curvature of the graph of $u_\rho$ in Minkowski spacetime $\mathbb{L}^{m+1}$. However, because of the lack of regularity of the functional when $\mid D\psi\mid = 1$, whether or not $u_\rho$ solves the Euler-Lagrange equation and how regular is $u_\rho$ are subtle issues that were investigated only for few classes of $\rho$. In this paper, we study both problems for general sources $\rho$, in a bounded domain with a Dirichlet boundary condition and in the entire $\mathbb{R}^m$. In particular, we give sufficient conditions to guarantee that $u_\rho$ solves Ethe uler-Lagrange equation and enjoys improved $W^{2,2}_{\textrm{loc}}$ estimates, and we construct examples helping to identify sharp thresholds for the regularity of $\rho$ to ensure the validity of the Euler-Lagrange equation. One of the main difficulties is the possible presence of light segments in the graph of $u_\rho$, which will be discussed in detail.