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

• Published in: Journal of mathematical physics 2011-02, Vol.52 (2), p.022902-022902-16
• Main Author: Kiessling, M. K.-H.
• Format: Article
• Language: English
• Subjects: Algebra; Convergence; Differential equations; Electromagnetics; Electromagnetism; Exact sciences and technology; Mathematical methods in physics; Mathematics; Physics; Sciences and techniques of general use
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.3548079

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 ∈ ( C 0 α ∩ L 1 ) ( R 3 ) and \documentclass[12pt]{minimal}\begin{document}$j\in (C^\alpha _0\cap L^1)({\mathbb R}^3)$\end{document} j ∈ ( C 0 α ∩ L 1 ) ( R 3 ) (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} C 0 1 , 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} C 0 1 , α size” of both, the Coulomb field generated by ρ and the Ampère field generated by j. No symmetry is assumed.