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The non-parametric problem

  • Tibor Radó
Part of the Ergebnisse der Mathematik und Ihrer Grenƶgebiete book series (volume 2)

Abstract

The problem of Plateau in the non-parametric form asks for a minimal surface bounded by a given curve and represented as a whole by an equation of the form z = z (x, y). The exact statement of the problem is as follows. Given, in the xy-plane, a Jordan curve Γ, and a continuous function φ(P) of the point P varying on r. Determine a function z (x, y) which is continuous in and on Γ, which has continuous derivatives of the first and second orders inside of Γ, which reduces to φ(P) on Γ, and which satisfies inside of Γ the partial differential equation
$$\left( {1 + {q^2}} \right)r - 2pqs + \left( {1 + {p^2}} \right)t = 0$$
, where
$$p = {z_x},\;q = {z_y},{\kern 1pt} \;r = {z_{xx}},\;s = {z_{xy}},\;t = {z_{yy}}$$
.

Keywords

Partial Derivative Variation Problem LIPSCHITZ Condition Jordan Curve Continuous Partial Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Hilbert: Über das Dirichletsche Prinzip. Jber. Deutsch. Math.-Vereinig. Vol. 8 (1900) pp. 184–188.MATHGoogle Scholar
  2. 2.
    Intégrale, longueur, aire. Ann. Mat. pura appl. Vol. 7 (1920) pp. 231–359.Google Scholar
  3. 1.
    Über die Variation der Doppelintegrale. J. reine angew. Math. Vol. 149 (1919) pp. 1–18.Google Scholar
  4. 2.
    Later on Haar generalized this lemma in several ways. See the expository-presentation by A. Haar: Zur Variationsrechnung. Abh. math. Semin. Hamburg. Univ. Vol. S (1930).Google Scholar
  5. 3.
    See for instance Bolza: Vorlesungen über Variationsrechnung, pp. 653–655.Google Scholar
  6. 4.
    Bemerkungen über das Prinzip der virtuellen Verrückungen. Ann. Soc. Polon. math. (1924).Google Scholar
  7. 5.
    Über die Umkehrung eines Satzes aus der Variationsrechnung. Acta Litt. Sci. Szeged Vol. 4 (1929) pp. 38–50.Google Scholar
  8. 6.
    Zur Variationsrechnung. Abh. math. Semin. Hamburg. Univ. Vol. 8 (1930).Google Scholar
  9. 7.
    See G. A. Bliss: Calculus of Variations (No. 1 of the Carus Mathematical Monographs).Google Scholar
  10. 1.
    Über die Umkehrung eines Satzes aus der Variationsrechnung. Acta Litt. Sci. Szeged Vol. 4 (1929) pp. 38–50.Google Scholar
  11. 1.
    J. Schauder: Über die Umkehrung eines Satzes aus der Variationsrechnung. Acta Litt. Sci. Szeged Vol. 4 (1929) pp. 38–50. — A. Haar: Zur Variationsrechnung. Abh. math. Semin. Hamburg. Univ. Vol. 8 (1930).Google Scholar
  12. 2.
    A. Haar: Über die Variation der Doppelintegrale. J. reine angew. Math. Vol. 149 (1919) pp. 1–18.MATHGoogle Scholar
  13. 1.
    See L. Lichtenstein: Neuere Entwicklung usw. Enzyklopädie der math. Wiss. Vol. 2 (3) pp. 1277–1334.Google Scholar
  14. 1.
    T. Radó: Über den analytischen Charakter der Minimalflächen. Math. Z. Vol. 24 (1925) pp. 321–327.MATHGoogle Scholar
  15. 2.
    Cf. II.11.Google Scholar
  16. 1.
  17. 1.
    T. Radó: Über den analytischen Charakter der Minimalflächen. Math. Z. Vol. 24 (1925) pp. 321–327Google Scholar
  18. 2.
    T. Radó: Bemerkung über die Differentialgleichungen zweidimensionale. Variationsprobleme. Acta Litt. Sci. Szeged Vol. 3 (1925) pp. 147–156.Google Scholar
  19. 1.
    T. Radó: Bemerkung über die Differentialgleichungen zweidimensionaler Yariation-probleme. Acta Litt. Sci. Szeged Vol. 3 (1925) pp. 147–156.Google Scholar
  20. 2.
    A. Haar: Über das Pi.ateausche Problem. Math. Ann. Vol. 97 (1927) pp 124–258.MathSciNetCrossRefGoogle Scholar
  21. 3.
    The theory of Junctions of two variables, satisfying the Lipschitz condition, has been the object of important investigations of H. Rademacher: Über partielle und totale Differenzierbarkeit I, II. Math. Ann. Vol. 70 (1919) pp. 340–359 andMathSciNetCrossRefGoogle Scholar
  22. 3a.
    H. Rademacher: Über partielle und totale Differenzierbarkeit I, II. Math. Ann. Vol. 81 (1920) pp. 52–63.MathSciNetCrossRefGoogle Scholar
  23. 1.
    See for references A. Haar: Über das Plateausche Problem. Math. Ann. Vol. 97 (1927) p. 127 andMathSciNetCrossRefGoogle Scholar
  24. 1a.
    A. Haar: Über das Plateausche Problem. Math. Ann. Vol. 97 (1927) p. 441.MathSciNetCrossRefGoogle Scholar
  25. 2.
    T. Radó: Geometrische Betrachtungen über zweidimensionale reguläre Variationsprobleme. Acta Litt. Sci. Szeged Vol. 2 (1926) pp. 228–253.MATHGoogle Scholar
  26. 3.
    J. v. Neumann: Über einen Satz der Variationsrechnung. Abh. math. Sem. Hamburg. Univ. Vol. 8 (1931) pp. 28–31.CrossRefGoogle Scholar
  27. 4.
    H. Lebesgue: Intégrale, longueur, aire. Ann. Mat. puraappl. Vol. 7 (1902) pp. 231–359.CrossRefGoogle Scholar
  28. 1.
    T. Radó: Über zweidimensionale reguläre Variationsprobleme. Math. Ann. Vol. 101 (1929) pp. 620–632. See in particular § 1, No. 2.MathSciNetMATHCrossRefGoogle Scholar
  29. 1.
    See, also for references, L. Tonelli: Sur la semi-continuité des intégrales doubles du Calcul des Variations. Acta math. Vol. 53 (1929) pp. 325–346 andMathSciNetMATHCrossRefGoogle Scholar
  30. 1a.
    J. E. McShane: On the semi-continuity of double integrals. Ann. of Math. Vol. 33 (1932) pp. 460–484.MathSciNetCrossRefGoogle Scholar
  31. 2.
    Über das Plateausche Problem. Math. Ann. Vol. 97 (1927) pp. 124–258.Google Scholar
  32. 3.
    T. Radó: Über zweidimensionale reguläre Variationsprobleme. Math. Ann. Vol. 101 (1929) pp. 620–632.MathSciNetMATHCrossRefGoogle Scholar
  33. 1.
    See for instance Pólya-Szegö: Aufgaben und Lehrsätze. Vol. 1 pp. 51–52, problems 70 and 71.Google Scholar
  34. 1.
    T. Radó: Über zweidimensionale reguläre Variationsprobleme. Math. Ann. Vol.101 (1929) pp. 620–632.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1933

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  • Tibor Radó

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