Constructing Graphs over ℝnwith Small Prescribed Mean-Curvature

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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 ℝn+1 (+ sign) and ℝ1,n (− sign) which are graphs {(x,u(x)):x∈ℝn of a smooth function u:ℝn→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 ℝ1,3, 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 languageEnglish (US)
Pages (from-to)1-25
Number of pages25
JournalMathematical Physics Analysis and Geometry
Issue number1
StatePublished - Dec 1 2015

All Science Journal Classification (ASJC) codes

  • Mathematical Physics
  • Geometry and Topology


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

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