Minimizers and Weak Solutions for Singular Born--Infeld Type Functionals

Abstract

We investigate the relation between minimizers and weak solutions for a class of singular functionals arising from Born--Infeld type theories $\mathcal{L}(s)$. In the setting of an electrostatic field $s=\frac{1}{2}|\nabla\phi|^2$, $\mathcal{L}(s)$ satisfies $\lim_{s\to(1/2)^-}\mathcal{L}(s)=+\infty$, which naturally enforces the finite gradient bound $|\nabla\phi|\le 1$, also called the truncation threshold. For a prescribed extended charge density $\rho$, we consider the relation between the weak solution of the system \begin{equation} \begin{cases} -{\rm div}\left(b\left(\frac12|\nabla\phi|^2\right)\nabla\phi\right)=\rho,& \text{in }\mathbb{R}^N,\\ b(s)=\mathcal{L}'(s),\quad\lim_{s\to\frac12^-} b(s)=+\infty,\\ \lim_{|x|\to\infty}\phi(x)=0 \end{cases} \end{equation} and the minimizer $\phi_0$ of the singular functional. We propose a monotonic approximation method to handle the intrinsic singularities of $\mathcal{L}(s)$. We prove that the gradient of the minimizer never touches the singular boundary $|\nabla\phi|^2=1$; this structural result yields a key integrability property, the existence and uniqueness of the minimizer, and the corresponding variational inequality. Under the additional assumption that $\rho$ is radially distributed, we show that the minimizer is the unique weak solution. Furthermore, we establish the $C^1$ and $C^2$ regularity of the minimizer under suitable integrability conditions on $\rho$, and provide a uniform estimate for the strict spacelikeness condition $|\nabla\phi_0|\le 1-\epsilon$, where the parameter $\epsilon>0$ is explicitly characterized in terms of the spatial dimension, the spatial region, and $\rho$. Our results extend the classical Born--Infeld theory to a general class of singular Born--Infeld type theories, thereby providing a unified framework for the variational analysis and regularity of such singular functionals and systems.