### Abstract

In this paper nonlinear Hodge theory and Banach algebra estimates are employed to construct a convergent series expansion which solves the prescribed mean curvature equation ± ∇ ⋅ (∇u/√1±| ∇u|^{2}) = nH for n-dimensional hypersurfaces in ℝ^{n+1} (+ sign) and ℝ^{1,n} (− sign) which are graphs {(x,u(x)):x∈ℝ^{n} of a smooth function u:ℝ^{n}→ℝ, and whose mean curvature function H is α-Hölder continuous and integrable, with small norm. The radius of convergence is estimated explicitly from below. Our approach is inspired by, and applied to, the Maxwell–Born–Infeld theory of electromagnetism in ℝ^{1,3}, for which our method yields the first systematic way of explicitly computing the electrostatic potential ϕ∝u for regular charge densities ρ∝H and small Born parameter, with explicit error estimates at any order of truncation of the series. In particular, our results level the ground for a controlled computation of Born–Infeld effects on the Hydrogen spectrum.

Original language | English (US) |
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Pages (from-to) | 1-25 |

Number of pages | 25 |

Journal | Mathematical Physics Analysis and Geometry |

Volume | 18 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2015 |

### All Science Journal Classification (ASJC) codes

- Mathematical Physics
- Geometry and Topology

### Keywords

- Classical field theory
- Convergent perturbation series
- Electromagnetism
- Prescribed mean curvature equation