Constructing graphs over $R^n$ with small prescribed mean-curvature

TL;DR

This paper develops a convergent series method to solve the prescribed mean curvature equation for hypersurfaces in Euclidean or Minkowskian space, enabling explicit computation of electrostatic potentials in a novel way.

Contribution

It introduces a new series expansion approach for solving the prescribed mean curvature equation in higher dimensions, inspired by electromagnetism theory.

Findings

Successfully constructs a convergent series solution for the mean curvature equation.

Provides the first explicit computational method for electrostatic potential with small Born parameter.

Applicable to hypersurfaces with bounded mean curvature in Euclidean and Minkowskian spaces.

Abstract

In this paper a convergent series expansion is constructed to solve the prescribed mean curvature equation for n-dimensional hypersurfaces in n+1 dimensional Euclidean or Minkowskian space(time) which are graphs of a smooth real function u, and whose mean curvature function H is not too large in Hoelder norm, and integrable. Our approach is inspired by the Maxwell-Born-Infeld theory of electromagnetism in Minkowski spacetime, for which our method yields the first systematic way of explicitly computing the electrostatic potential u for regular charge densities proportional to H and small Born parameter.