The Mathematical Intelligencer

, Volume 11, Issue 4, pp 50–60 | Cite as

The calculus of variations today

  • Stefan Hildebrandt


Weak Solution Minimal Surface Morse Theory Soap Film Nonlinear Elliptic System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints.Memoirs of the Amer. Math. Soc. 4, No. 165 (1976).Google Scholar
  2. 2.
    J. M. Ball, editor,Systems of nonlinear partial differential equations. NATO ASI series, Series C, No. 111, Proc. of NATO Adv. Study Inst. Oxford 1982, Reidel Publ., Dordrecht-Boston-Lancaster (1983).MATHGoogle Scholar
  3. 3.
    M. Giaquinta,Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Math. Studies, Princeton Univ. Press (1983).Google Scholar
  4. 4.
    D. Gilbarg and N. Trudinger,Elliptic partial differential equations of second order, Heidelberg-New York: Springer- Verlag (1977).CrossRefMATHGoogle Scholar
  5. 5.
    D. Hilbert, Mathematische Probleme,Archiv f. Math. und Phys. 3. Reihe, Bd. 1, 44–63, 213-237 (1901), und: Ges.Abhandl. Bd. 3, Springer-Verlag, 290-329 (1935).MATHGoogle Scholar
  6. 6.
    S. Hildebrandt, Nonlinear elliptic systems and harmonic mappings,Proc. 1980 Beijing Symp. on Diff. Geom. and Diff. Equ., Vol. 1, Beijing: Science Press (1982) 481–615.Google Scholar
  7. 7.
    S. Hildebrandt, Calculus of Variations Today, reflected in the Oberwolfach meetings,Perspectives in mathematics, W. Jäger, J. Moser, R. Remmert, (ed.), Basel-Boston-Stuttgart: Birkhäuser Verlag (1983), 321–336.Google Scholar
  8. 8.
    S. Hildebrandt and A. Tromba,Mathematics and Optimal Form, Scientif. Amer. Library, New York: W. H. Freeman and Co. (1985).Google Scholar
  9. 9.
    O. A. Ladyzenskaja and N. N. Uralzeva,Linear and quasilinear elliptic equations, New York and London: Acad. Press (1968).Google Scholar
  10. 10.
    C. B. Morrey,Multiple integrals in the calculus of variations, Berlin-Heidelberg-New York: Springer-Verlag (1966).MATHGoogle Scholar
  11. 11.
    J. C. C. Nitsche,Vorlesungen über Minimalflächen, Berlin- Heidelberg-New York: Springer-Verlag (1975).CrossRefMATHGoogle Scholar
  12. 12.
    L. Simon,Lectures on Geometric Measure Theory, Centre for Math. Anal., Austral. Nat. Univ. Canberra Vol. 3 (1983).Google Scholar

Picture Credits

  1. Figures 1, 6, 10: S. Hildebrandt and A. Tromba,Mathematics and Optimal Form, New York: W. H. Freeman & Co. (1985).Google Scholar
  2. Figure 2: S. Hildebrandt and J. C. C. Nitsche, A uniqueness theorem for surfaces of least area with partially free boundaries on obstacles,Archive for Rational Mechanics and Analysis 79, 189–218 (1982).CrossRefMATHMathSciNetGoogle Scholar
  3. Figures 3, 5, 8, 9: Bildarchiv. Inst. für Leichte Flächentragwerke, Universität Stuttgart.Google Scholar
  4. Figure 7: E. Häckel,Reports of the Scientific Results of H.M.S. Challenger, London, 1881–89.Google Scholar
  5. Figure 14: B. Winkel und J. Krön,Minimalwegenetze mit vielen Knoten, Studienarbeit 2/1985, Inst. für Leichte Flächentragwerke, TU Stuttgart.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 1989

Authors and Affiliations

  • Stefan Hildebrandt
    • 1
  1. 1.Mathematisches InstitutUniversität BonnBonnFederal Republic of Germany

Personalised recommendations