2026/07/20 by Tengyang Liu, Ruifeng Zhang
#math.AP #math-ph #math.MP
We investigate the relation between minimizers and weak solutions for a class of singular functionals arising from Born--Infeld type theories L(s). In the setting of an electrostatic field s=(1)/(2)|∇ϕ|2, L(s) satisfies lims→(1/2)-L(s)=+∞, which naturally enforces the finite gradient bound |∇ϕ|≤ 1, also called the truncation threshold. For a prescribed extended charge density ρ, we consider the relation between the weak solution of the system \begincases -\rm div(b(\frac12|∇ϕ|2)∇ϕ)=ρ, in ℝN,
b(s)=L'(s), lims→\frac12- b(s)=+∞,
lim|x|→∞ϕ(x)=0 \endcases and the minimizer ϕ0 of the singular functional. We propose a monotonic approximation method to handle the intrinsic singularities of L(s). We prove that the gradient of the minimizer never touches the singular boundary |∇ϕ|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 ρ is radially distributed, we show that the minimizer is the unique weak solution. Furthermore, we establish the C1 and C2 regularity of the minimizer under suitable integrability conditions on ρ, and provide a uniform estimate for the strict spacelikeness condition |∇ϕ0|≤ 1-ε, where the parameter ε>0 is explicitly characterized in terms of the spatial dimension, the spatial region, and ρ. 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.