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C1,α Theory for the Prescribed Mean Curvature Equation with Dirichlet Data

Abstract

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. With such a solution we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a C 1,α manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class C 1,α.

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Correspondence to Theodora Bourni.

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Communicated by John M. Lee.

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Bourni, T. C1,α Theory for the Prescribed Mean Curvature Equation with Dirichlet Data. J Geom Anal 21, 982–1035 (2011). https://doi.org/10.1007/s12220-010-9176-6

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  • DOI: https://doi.org/10.1007/s12220-010-9176-6

Keywords

  • Prescribed mean curvature equation
  • Minimal surfaces
  • Boundary regularity

Mathematics Subject Classification (2000)

  • 53A10
  • 49Q15