Convergent perturbative power series solution of the stationary Maxwell–Born–Infeld field equations with regular sources

Abstract

The stationary Maxwell–Born–Infeld field equations of electromagnetism with regular sources \documentclass[12pt]{minimal}\begin{document}$\rho \in (C^\alpha _0\cap L^1)({\mathbb R}^3)$\end{document}ρ∈(C0α∩L1)(R3) and \documentclass[12pt]{minimal}\begin{document}$j\in (C^\alpha _0\cap L^1)({\mathbb R}^3)$\end{document}j∈(C0α∩L1)(R3) (componentwise) are solved using a perturbation series expansion in powers of Born's electromagnetic constant. The convergence in \documentclass[12pt]{minimal}\begin{document}$C^{1,\alpha }_0$\end{document}C01,α of the power series for the fields is proved with the help of Banach algebra arguments and complex analysis. The finite radius of convergence depends on the “\documentclass[12pt]{minimal}\begin{document}$C^{1,\alpha }_0$\end{document}C01,α size” of both, the Coulomb field generated by ρ and the Ampère field generated by j. No symmetry is assumed.