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On a Minimal Hypersurface in \(\mathbb {R}^{4}\)

  • Ammar KhanferEmail author
  • Kirk E. Lancaster
Article
  • 28 Downloads

Abstract

For a nonparametric prescribed mean curvature surface \(z=f(x,y)\) in a vertical cylinder \(\Omega \times \mathbb {R}\subset \mathbb {R}^{3},\) the existence and behavior of the radial limits of f at a nonconvex corner \(\mathcal{O}_{2}\in \partial \Omega \) are well studied and understood. For a nonparametric prescribed mean curvature hypersurface \(z=f(\mathbf{x})\) in a vertical cylinder \(\Omega \times \mathbb {R}\subset \mathbb {R}^{n+1},\)\(n\ge 3,\) at a nonconvex corner \(\mathcal{O}_{n}\in \partial \Omega ,\) the boundary behavior of f is largely unknown. We shall consider the specific example of a nonparametric minimal hypersurface in \(\mathbb {R}^{4}\) and investigate its behavior at a nonconvex conical point of the boundary of its domain.

Keywords

Minimal hypersurface Dirichlet problem Variational solution Conical point Radial limits 

Notes

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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Imam Mohammad Ibn Saud Islamic UniversityRiyadhSaudi Arabia
  2. 2.WichitaUSA

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