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

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.

28 pages, LaTeX, To appear in Mathematical Physics, Analysis, and Geometry (2015)

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Cited by in corpus (3)

• Some uniqueness results for stationary solutions to the Maxwell-Born-Infeld field equations and their physical consequences
• On the Schrödinger spectrum of a hydrogen atom with electrostatic Bopp-Landé-Thomas-Podolsky interaction between electron and proton
• Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model