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A geometric maximum principle, plateau’s problem for surfaces of prescribed mean curvature, and the two dimensional analogue of the catenary

Part of the Lecture Notes in Mathematics book series (LNM,volume 1357)

Keywords

  • Euler Equation
  • Variational Problem
  • Minimal Surface
  • Branch Point
  • Principal Curvature

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References

  1. Bemelmans, J., Dierkes, U.: On a singular variational integral with linear growth, I: existence and regularity of minimizers. Arch. Rat. Mech. Analysis, 100, 83–103 (1987).

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Böhme, R., Hildebrandt, S., Tausch, E.: The two dimensional analogue of the catenary. Pacific Journal Math 88, 247–278 (1980).

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Dierkes, U.: Singuläre Variationsprobleme und Hindernisprobleme. Dissertation Bonn 1984, Bonner Math. Schriften 155.

    Google Scholar 

  4. Dierkes, U.: Plateau’s problem for surfaces of prescribed mean curvature in given regions. manuscripta math. 56, 313–331 (1986).

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Dierkes, U.: A geometric maximum principle for surfaces of prescribed mean curvature in Riemannian manifolds. To appear in Zeitschrift für Analysis und ihre Anwendungen.

    Google Scholar 

  6. Gromoll, D., Klingenberg, W., Meyer, W.: Riemannsche Geometrie im Großen. Berlin, Heidelberg, New York, Springer 1968.

    CrossRef  MATH  Google Scholar 

  7. Grüter, M.: Regularity of weak H-surfaces. Journ. Reine Angew. Math. 329 (1981), 1–15.

    MathSciNet  MATH  Google Scholar 

  8. Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. Ann. Math. 97, 1 275–305 (1973).

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Hartmann, P., Wintner A.: On the local behavior of solutions of non-parametric partial differential equations. Am. J. Math. 75, 149–476 (1953).

    Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Hildebrandt, S.: Maximum principle for minimal surfaces and for surfaces of continuous mean curvature. Math. Z. 128, 253–269.

    Google Scholar 

  14. Hildebrandt, S.: On the regularity of solutions of two-dimensional variational problems with obstructions. CPAM 25, 479–496 (1972).

    MathSciNet  MATH  Google Scholar 

  15. Hildebrandt, S.: Interior C1+α-regularity of solutions of two dimensional variational problems with obstacles. Math. Z. 131, 233–240 (1973).

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Hildebrandt, S.: Einige Bemerkungen über Flächen beschränkter mittlerer Krümmung. Math. Z. 115, 169–178 (1971).

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  20. Kaul, H: Isoperimetrische Ungleichung und Gauss-Bonnet Formel für H-Flächen in Riemannschen Mannigfaltigkeiten. Arch. Rat. Mech. Analysis 45 (1972), 194–221.

    CrossRef  MathSciNet  Google Scholar 

  21. Keiper, J.: The axially symmetric n-tectum. Preprint, Toledo University (1980).

    Google Scholar 

  22. Kneser, H.: Lösung der Aufgabe 41. Jahresbericht der DMV 35, 123–124 (1926).

    Google Scholar 

  23. Michael, J.H., Simon, L.M.: Sobolev and mean-value inequalities on generalized submanifolds of ℝn. CPAM 26 (1973), 361–379.

    MathSciNet  Google Scholar 

  24. Nitsche, J.C.C.: Vorlesungen über Minimalflächen. Springer Grundlehren 199, Berlin-Heidelberg-New York 1975.

    Google Scholar 

  25. Radó, T.: The problem of least area and the problem of Plateau. Math. Z. 32 (1930), 763–796.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Steffen, K.: Ein verbesserter Existenzsatz für Flächen konstanter mittlerer Krümmung. manuscripta math. 6, 105–139 (1972).

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. Wente, H.: An existence theorem for surfaces of constant mean curvature. J. math. Analysis Appl. 26, 318–344 (1969).

    CrossRef  MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  MATH  Google Scholar 

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Dierkes, U. (1988). A geometric maximum principle, plateau’s problem for surfaces of prescribed mean curvature, and the two dimensional analogue of the catenary. In: Hildebrandt, S., Leis, R. (eds) Partial Differential Equations and Calculus of Variations. Lecture Notes in Mathematics, vol 1357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082864

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  • DOI: https://doi.org/10.1007/BFb0082864

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