manuscripta mathematica

, Volume 28, Issue 1–3, pp 185–206 | Cite as

On variational problems with obstacles and integral constraints for vector-valued functions

  • Stefan Hildebrandt
  • Michael Meier


The authors prove existence and regularity for vectorvalued solutions of n-dimensional variational problems with boundary conditions, integral constraints, and obstacles as side conditions. Main emphasis is given to the regularity proof in the case n=2 generalizing a well known technique due to C. B. Morrey. In addition, a regularity result is stated for the general n-dimensional case.


Boundary Condition Number Theory Variational Problem Algebraic Geometry Topological Group 
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]
    BERKOVITZ, L. D.: Lower Semicontinuity of Integral Functionals. Transactions of the American Mathematical Society,192, 1–57 (1974)Google Scholar
  2. [2]
    DE GIORGI, E.: Teoremi di semicontinuitá nel calcolo delle variazioni. Istituto Nazionale di Alta Matematica Roma, Lezioni tenute nell' anno academico 1968–69Google Scholar
  3. [3]
    EISEN, G.: A Selection Lemma for Sequences of Measurable Sets, and Lower Semicontinuity of Multiple Integrals. To appear in Manuscripta Math.Google Scholar
  4. [4]
    FREHSE, J.: On systems of Second-Order Variational Inequalities. Israel J. Math.15, 421–429 (1973)Google Scholar
  5. [5]
    HILDEBRANDT, S., and 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
  6. [6]
    HILDEBRANDT, S.: On the Regularity of Solutions of Two-dimensional Variational Problems with Obstructions. Commun. Pure Appl. Math.25, 479–496 (1972)Google Scholar
  7. [7]
    HILDEBRANDT, S.: Interior C1+α-Regularity of Two-Dimensional Variational Problems with Obstacles. Math. Z.131, 233–240 (1973)Google Scholar
  8. [8]
    HILDEBRANDT, S., and WENTE, H. C.: Variational Problems with Obstacles and a volume Constraint Math. Z.135, 55–68 (1973)Google Scholar
  9. [9]
    HILDEBRANDT, S., and WIDMAN, K.-O.: Some Regularity Results for Quasi-Linear Elliptic Systems of Second Order. Math. Z.142, 67–86 (1975)Google Scholar
  10. [10]
    HILDEBRANDT, S., and WIDMAN, K.-O.: On the Hölder Continuity of Quasi-Linear Elliptic Systems of Second Order. Ann. Scuola Norm. Sup. Pisa (Ser. IV),4, 145–178 (1977)Google Scholar
  11. [11]
    HILDEBRANDT, S., and WIDMAN, K.-O.: Variational Inequalities for Vector Valued Functions. To appear in Journ. f. Reine und Angewandte Math.Google Scholar
  12. [12]
    MORREY, C. B., Jr.: Existence and Differentiability Theorems for the Solutions of Variational Problems for Multiple Integrals. Bull. Amer. Math. Soc.46, 439–458 (1940)Google Scholar
  13. [13]
    MORREY, C. B., Jr.: Multiple Integral Problems in the Calculus of Variations and Related Topics. Univ. of California Publ. in Math., new ser. I, 1–130 (1943)Google Scholar
  14. [14]
    MORREY, C. B., Jr.: Quasi-Convexity and the Lower Semicontinuity of Multiple Integrals. Pacific J. Math.2, 25–53 (1952)Google Scholar
  15. [15]
    MORREY, C. B., Jr.: Second Order Elliptic Systems of Differential Equations. Ann. of Math. Studies No. 33, Princeton Univ. Press, 101–159 (1954)Google Scholar
  16. [16]
    MORREY C. B., Jr.: Multiple Integral Problems in the Calculus of Variations and Related Topics. Ann. Scuola Norm. Sup. Pisa (III)14, 1–61 (1960)Google Scholar
  17. [17]
    MORREY, C. B., Jr.: Multiple Integrals in the Calculus of Variations. Die Grundlehren der mathematischen Wissenschaften 130. Berlin-Heidelberg-New York: Springer 1966Google Scholar
  18. [18]
    RIDDELL, R. C.: Eigenvalue Problems for Nonlinear Elliptic Variational Inequalities in a Cone. J. Functional Anal.26, 333–355 (1977)Google Scholar
  19. [19]
    STEFFEN, K.: Flächen konstanter mittlerer Krümmung mit vorgegebenem Volumen oder Flächeninhalt. Arch. Rat. Mech. Analysis49, 99–128 (1972)Google Scholar
  20. [20]
    TOMI, F.: Variationsprobleme vom Dirichlet-Typ mit einer Ungleichung als Nebenbedingung. Math. Z.128, 43–74 (1972)Google Scholar
  21. [21]
    WENTE, H. C.: A General Existence Theorem for Surfaces of Constant Mean Curvature. Math. Z.120, 277–288 (1971)Google Scholar
  22. [22]
    WIEGNER, M.: Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme. Math. Z.147, 21–28 (1976)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Stefan Hildebrandt
    • 1
  • Michael Meier
    • 1
  1. 1.Mathematisches InstitutUniversität BonnBonn 1Bundesrepublik Deutschland

Personalised recommendations