Between Maxwell and Born-Infeld: the presence of the magnetic field

TL;DR

The paper develops a variational framework for an interpolated electromagnetic Lagrangian density $\mathcal{L}_q$ with $q\in[1,2]$ that connects Born--Infeld theory to Maxwell theory. It analyzes two static settings: an electrostatic problem in the presence of a fixed magnetic field $B$ and a magnetostatic problem with an external current $J$. For the electrostatic case, it proves that the energy functional $I_B$ on the convex set $X_B$ attains a unique minimizer $\phi_\rho$ and yields a weak, radial solution when $\rho$ is radial and $B$ has appropriate symmetry; for the magnetostatic problem, it introduces a divergence-free space $\mathcal{A}$ and shows the energy $\mathcal{J}$ is strictly convex and coercive on $\mathcal{A}$ for $q\in(6/5,2)$, yielding a unique nontrivial cylindrically symmetric minimizer of the form $A(x)=\frac{u(r,x_3)}{r}(-x_2,x_1,0)$. The combination of radial and cylindrical symmetry constraints, together with the reflexivity induced by $q>6/5$, yields rigorous existence results that bridge the two theories while ensuring finite energy for point charges.

Abstract

Our motivation is to consider an electromagnetic Lagrangian density $\mathcal{L}_q$, depending on a parameter such that, for $q=1$ it corresponds to the Born-Infeld Lagrangian density and for $q=2$ it restores the Maxwell one. The model in the presence of given charge and current densities is investigated. We solve the pure magnetostatic problem for $q\in(6/5,2)$. We also study the electrostatic problem in the presence of an assigned magnetic field for $q\in[1,2)$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Proposition 2.4
• proof
• Theorem 2.5