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Area minimizing hypersurfaces with prescribed volume and boundary

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References

  • [All 1] Allard, W.K.: On the first variation of a varifold. Ann. Math.95, 417–491 (1972)

    Article  MathSciNet  Google Scholar 

  • [All 2] Allard, W.K.: On the first variation of a varifold: boundary behaviour. Ann. Math.101, 418–446 (1975)

    Article  MathSciNet  Google Scholar 

  • [Alm 1] Almgren, F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc.165 (1976)

  • [Alm 2] Almgren, F.J.: Optimal Isoperimetric Inequalities: Indiana Univ. Math. J.35, 451–547 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • [BC] Brezis, H., Coron, J.-M.: Multiple solutions of H-systems and Rellich's conjecture. Commun. Pure Appl. Math.37, 149–187 (1984)

    MATH  MathSciNet  Google Scholar 

  • [DF 1] Duzaar, F., Fuchs, M.: On integral currents with constant mean curvature. Rend. Semin. Mat. Univ. Padova85, 79–103 (1991)

    MATH  MathSciNet  Google Scholar 

  • [DF2] Duzaar, F., Fuchs, M.: A general existence theorem for integral currents with prescribed mean curvature form. Boll. Unione Mat. Ital. (to appear)

  • [DF 3] Duzaar, F., Fuchs, M.: On the existence of integral currents with prescribed mean curvature vector. Manuscr. Math.67, 41–67 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  • [DS] Duzaar, F., Steffen, K.: A partial regularity theorem for harmonic maps at a free boundary. Asymptotic Anal.2, 299–343 (1989)

    MATH  MathSciNet  Google Scholar 

  • [Fe] Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969

    MATH  Google Scholar 

  • [Fe 1] Federer, H.: The singular set of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Am. Math. Soc.76, 767–771 (1979)

    Article  MathSciNet  Google Scholar 

  • [He] Heinz, E.: On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch. Ration. Mech. Anal.35, 249–252 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  • [HS] Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. Math.110, 439–486 (1979)

    Article  MathSciNet  Google Scholar 

  • [Gi] Giusti, E.: Minmal surfaces and functions of bounded variation. (Monogr. Math., Basel) Basel Boston Stuttgart: Birkhäuser 1984

    Google Scholar 

  • [GG] Gonzalez, E.H.A., Greco, G.H.: Una nuova dimonstrazione della proprietà isoperimetrica dell'ipersfera nella classe degli insiemi aventi perimetro finito. Ann. Univ. Ferrara23, 251–256 (1977)

    MathSciNet  MATH  Google Scholar 

  • [GMT] Gonzalez, E., Massari, U., Tamanini, I.: On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J.32, 25–37 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  • [GT] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, second edition. Berlin Heidelberg New York: Springer 1977

    MATH  Google Scholar 

  • [Gu] Gulliver, R.: On the nonexistence of a hypersurface of prescribed mean curvature with given boundary. Manuscr. Math.11, 15–39 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  • [MM] Massari, U., Miranda, M.: Minimal surfaces of codimension one. (Math. Stud., vol. 91) Amsterdam: North Holland 1984

    MATH  Google Scholar 

  • [Mo] Morrey, C.B.: Second order elliptic systems of differential equations (Ann. Math. Stud., vol. 33, pp. 101–159) Princeton: Princeton University Press 1954

    Google Scholar 

  • [Si] Simon, L.: Lectures on geometric measure theory: Canberra: Aust. Natl. Univ. 1984 (Proc. Cent. Math. Anal. Aust. Natl. Univ., vol. 3)

    Google Scholar 

  • [Ste 1] Steffen, K.: Flächen vorgeschriebener mittlerer Krümmung mit vorgegebenem Volumen oder Flächeninhalt. Arch. Ration. Mech. Anal.49, 191–217 (1972)

    Article  MathSciNet  Google Scholar 

  • [Ste 2] Steffen, K.: On the nonuniqueness of surfaces with prescribed mean curvature spanning a given contour. Arch. Ration. Mech. Anal.94, 101–122 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • [Str 1] Struwe, M.: Nonuniqueness in the Plateau-problem for surfaces of constant mean curvature. Arch. Ration. Mech. Anal.93, 135–157 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • [Str 2] Struwe, M.: Large H-surfaces via the mountain-pass-lemma. Math. Ann.270, 441–459 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • [We 1] Wente, H.: A general existence theorem for surfaces of constant mean curvature. Math. Z.120, 277–288 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  • [We 2] Wente, H.: The Dirichlet problem with a volume constraint. Manuscr. Mat.11, 141–157 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  • [We 3] Wente, H.: Large solutions to the volume constrained Plateau problem. Arch. Ration. Mech. Anal.75, 59–77 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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Duzaar, F., Steffen, K. Area minimizing hypersurfaces with prescribed volume and boundary. Math Z 209, 581–618 (1992). https://doi.org/10.1007/BF02570855

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