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On the nonuniqueness of surfaces with constant mean curvature spanning a given contour

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References

  1. Struwe, M.: Nonuniqueness in the Plateau problem for surfaces of constant mean curvature. Arch. Rational Mech. Anal. 93, 135–157 (1986).

    Google Scholar 

  2. Alt, H. W.: Verzweigungspunkte von H-Flächen I, II. Math. Z. 127, 333–362 (1972). Math. Ann. 201, 33–56 (1973).

    Google Scholar 

  3. Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. of Math. 97 275–305 (1973).

    Google Scholar 

  4. Gulliver, R., Ossermann, R., Royden, H. L.: A theory of branched immersions of surfaces. Amer. J. Math. 95, 750–812 (1973).

    Google Scholar 

  5. Gulliver, R., Spruck, J.: The Plateau problem for surfaces of prescribed mean curvature in a cylinder. Inventiones Math. 13, 169–178 (1971).

    Google Scholar 

  6. Gulliver, R., Spruck, J.: Existence theorems for parametric surfaces of prescribed mean curvature. Indiana Univ. Math. J. 22, 445–472 (1972).

    Google Scholar 

  7. Heinz, E.: Über die Existenz einer Fläche konstanter mittlerer Krümmung bei vorgegebener Berandung. Math. Ann. 127, 258–287 (1954).

    Google Scholar 

  8. Heinz, E.: On the existence problem for surfaces of constant mean curvature. Comm. Pure Appl. Math. 9, 467–470 (1956).

    Google Scholar 

  9. Heinz, E.: Ein Regularitätssatz für Flächen beschränkter mittlerer Krümmung. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl., Jahrgang 1969, 107–118.

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

    Google Scholar 

  11. Heinz, E.: Über das Randverhalten quasilinearer elliptischer Systeme mit isothermen Parametern. Math. Z. 113, 99–105 (1970).

    Google Scholar 

  12. Heinz, E., Tomi, F.: Zu einem Satz von Hildebrandt über das Randverhalten von Minimalflächen. Math. Z. 111, 372–386 (1969).

    Google Scholar 

  13. Hildebrandt, S.: Über Flächen konstanter mittlerer Krümmung. Math. Z. 112, 107–144 (1969).

    Google Scholar 

  14. Hildebrandt, S.: Randwertprobleme für Flächen mit vorgeschriebener mittlerer Krümmung und Anwendungen auf die Kapillaritätstheorie I. Math. Z. 112, 205–213 (1969).

    Google Scholar 

  15. Hildebrandt, S.: On the Plateau problem for surfaces of constant mean curvature. Comm. Pure Appl. Math. 23, 97–114 (1970).

    Google Scholar 

  16. Hildebrandt, S.: Über einen neuen Existenzsatz für Flächen vorgeschriebener mittlerer Krümmung. Math. Z. 119, 267–272 (1971).

    Google Scholar 

  17. Hildebrandt, S.: Maximum principles for minimal surfaces and for surfaces of continuous mean curvature. Math. Z. 128, 253–269 (1972).

    Google Scholar 

  18. Hildebrandt, S.: Nonlinear elliptic systems and harmonic mappings. SFB lecture notes no. 3, Bonn University 1980. Proceedings of Beijing Symposium on Differential Geometry and Differential Equations, Science Press, Beijing, 1982.

    Google Scholar 

  19. Hildebrandt, S., Kaul, H.: Two-dimensional variational problems with obstructions, and Plateau's problem for H-surfaces in a Riemannian manifold. Comm. Pure Appl. Math. 25, 187–223 (1972).

    Google Scholar 

  20. Hildebrandt, S., Widman, K.-O.: Variational inequalities for vector valued functions. J. Reine Angew. Math. 309, 191–220 (1979).

    Google Scholar 

  21. Jäger, W.: Ein Maximumprinzip für ein System nichtlinearer Differentialgleichungen. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Klasse II, Jahrgang 1976, 157–164.

  22. Morrey, C. B.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966.

    Google Scholar 

  23. Osserman, R.: A proof of the regularity everywhere of the classical solution to Plateau's problem. Ann. of Math. 91, 550–569 (1970).

    Google Scholar 

  24. Serrin, J.: On surfaces of constant mean curvature that span a given space curve. Math. Z. 112, 77–88 (1969).

    Google Scholar 

  25. Steffen, K.: Ein verbesserter Existenzssatz für Flächen konstanter mittlerer Krümmung. Manuscripta Math. 6, 105–139 (1972).

    Google Scholar 

  26. Steffen, K.: Flächen konstanter mittlerer Krümmung mit vorgegebenem Volumen oder Flächeninhalt. Arch. Rational Mech. Anal. 49, 99–128 (1972).

    Google Scholar 

  27. Steffen, K.: Isoperimetric inequalities and the problem of Plateau. Math. Ann. 222, 97–144 (1976).

    Google Scholar 

  28. Steffen, K.: On the existence of surfaces with prescribed mean curvature and boundary. Math. Z. 146, 113–135 (1976).

    Google Scholar 

  29. Steffen, K., Wente, H.: The non-existence of branch points in solutions to certain classes of Plateau type variational problems. Math. Z. 163, 211–238 (1978).

    Google Scholar 

  30. Tomi, F.: Ein einfacher Beweis eines Regularitätssatzes für schwache Lösungen gewisser elliptischer Systeme. Math. Z. 112, 214–218 (1969).

    Google Scholar 

  31. Wente, H. C.: An existence theorem for surfaces of constant mean curvature. J. Math. Anal. Appl. 1, 318–344 (1969).

    Google Scholar 

  32. Wente, H. C.: A general existence theorem for surfaces of constant mean curvature. Math. Z. 120, 277–288 (1971).

    Google Scholar 

  33. Wente, H. C.: The Dirichlet problem with a volume constraint. Manuscripta Math. 11, 141–157 (1974).

    Google Scholar 

  34. Wente, H. C.: The differential equation NETH with vanishing boundary values. Proc. A.M.S. 50, 131–137 (1975).

    Google Scholar 

  35. Wente, H. C.: Large solutions to the volume constrained Plateau problem. Arch. Rational Mech. Anal. 75, 59–77 (1980).

    Google Scholar 

  36. Werner, H.: Das Problem von Douglas für Flächen konstanter mittlerer Krümmung. Math. Ann. 133, 303–319 (1957).

    Google Scholar 

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  1. Mathematisches Institut, Universität Düsseldorf, Universitätsstraße, 1, Düsseldorf

    Klaus Steffen

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  1. Klaus Steffen
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Communicated by J. C. C. Nitsche

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Steffen, K. On the nonuniqueness of surfaces with constant mean curvature spanning a given contour. Arch. Rational Mech. Anal. 94, 101–122 (1986). https://doi.org/10.1007/BF00280429

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