Constructing Graphs over ℝⁿwith Small Prescribed Mean-Curvature

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 ℝ^1,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 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.