manuscripta mathematica

, Volume 72, Issue 1, pp 131–140 | Cite as

Hypersurfaces of prescribed mean curvature enclosing a given body

  • Martin Fuchs


Given a smooth domain Ω in ℝ m+1 with compact closure and a smooth integrable functionh: ℝ m+1→ℝ satisfyingh(x)H ∂Ω (x) on ∂Ω whereH ∂ω denotes the mean curvature of ∂Ω calculated w.r.t. the interior unit normal we show that there is a setA⊂ℝ m+1 with the properties\(A \supset \bar \Omega \) andH ∂A=h on ∂A.

Key words

Caccioppoli sets mean curvature closed hypersurfaces 


49Q20 49F20 49F22 53A10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F.J. Almgren, Optimal isoperimetric inequalities. Indiana Univ., Mat. J. 35 (1986), 451–547zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    F. Duzaar, Ströme vorgeschriebener mittlerer Krümmung-Existenz und Regularität. Habilitationsschrift Universität Düsseldorf, Düsseldorf 1990Google Scholar
  3. [3]
    E. DeGiorgi, Frontiere orientate di misura minima. Sem. Mat. Scuola Norm. Sup. Pisa 1960–61Google Scholar
  4. [4]
    F. Duzaar, M. Fuchs, On the existence of integral currents with prescribed mean curvature vector. manus. math. 67 (1990), 41–67zbMATHMathSciNetGoogle Scholar
  5. [5]
    F. Duzaar, M. Fuchs, On integral currents with constant mean curvature. Preprint No. 90 SFB 256, U Bonn (to appear in Rend. Sem. Mat. Univ. Padova 1991)Google Scholar
  6. [6]
    F. Duzaar, F. Fuchs, A general existence theorem for integral currents with prescribed mean curvature form. Preprint 1990Google Scholar
  7. [7]
    H. Federer, Geometric measure theory. Springer Verlag 1969Google Scholar
  8. [8]
    E. Guisti, Minimal surfaces and functions of bounded variation. Birkhäuser Monographs in Mathematics 1984Google Scholar
  9. [9]
    L. Simon, Lectures on geometric measure theory. Proceedings C.M.A.3, Canberra 1983Google Scholar
  10. [10]
    K. Steffen, On the existence of surfaces with prescribed mean curvature and boundary. Math. Z. 146, 113–135 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    I. Tamanini, Regularity results for almost minimal oriented hypersurfaces. Quaderni Del Departimento di Matematica del Univ. Di Lecce Q1 1984Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Martin Fuchs
    • 1
  1. 1.Technische Hochschule DarmstadtDarmstadt

Personalised recommendations