On the regularity of the minimizer of the electrostatic Born-Infeld energy

TL;DR

This paper investigates the regularity of the minimizer of the electrostatic Born-Infeld energy functional, proving higher regularity under certain conditions on the charge density and smallness assumptions.

Contribution

It establishes the regularity of the minimizer in Sobolev spaces and shows that for small charge densities, the minimizer solves the associated PDE with improved smoothness.

Findings

Minimizer belongs to W_{loc}^{2,2} under certain integrability conditions.

For small charge densities, the minimizer is a weak solution of the PDE.

The minimizer is locally C^{1,α} smooth for some α in (0,1).

Abstract

We consider the electrostatic Born-Infeld energy \begin{equation*} \int_{\mathbb{R}^N}\left(1-{\sqrt{1-|\nabla u|^2}}\right)\, dx -\int_{\mathbb{R}^N}\rho u\, dx, \end{equation*} where $\rho \in L^{m}(\mathbb{R}^N)$ is an assigned charge density, $m \in [1,2_*]$, $2_*:=\frac{2N}{N+2}$, $N\geq 3$. We prove that if $\rho \in L^q(\mathbb{R}^N) $ for $q>2N$, the unique minimizer $u_\rho$ is of class $W_{loc}^{2,2}(\mathbb{R}^N)$. Moreover, if the norm of $\rho$ is sufficiently small, the minimizer is a weak solution of the associated PDE \begin{equation}\label{eq:BI-abs} \tag{$\mathcal{BI}$} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \rho \quad\hbox{in }\mathbb{R}^N, \end{equation} with the boundary condition $\lim_{|x|\to\infty}u(x)=0$ and it is of class $C^{1,\alpha}_{loc}(\mathbb{R}^N)$, for some $\alpha \in (0,1)$.