Using the concept of well-posedness under perturbations we give an answer to the following question posed by Stanislaw Ulam : “when it is true that solutions of two problems in the calculus of variations which corresponds to “close” physical data must be close to each other?”(“A collection of mathematical problems”, 1960).


well-posedness integral functionals bounded Hausdorff convergence 


  1. [1]
    H. Attouch and R. Wets. Quantitative stability of variational systems. I. The epigraphical distance. Trans. Amer. Math. Soc., 328(2):695–729, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    H. Attouch and R. Wets. Quantitative stability of variational systems. II. A framework for nonlinear conditioning. SIAM J. Optim., 3(2):359–381, 1993.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    G. Beer. Topologies on closed and closed convex sets, volume 268 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1993.Google Scholar
  4. [4]
    N. A. Bobylëv. On a problem of s. ulam. Nonlinear Anal., 24(3):309–322, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Z. Chbani. Caractérisation de la convergence d’intégrates définies à partir d’opérateurs elliptiques par la convergence des coefficients. Sém. Anal. Convexe, 21:Exp. No. 12, 14, 1991.Google Scholar
  6. [6]
    A. L. Dontchev and T. Zolezzi. Well-posed optimization problems, volume 1543 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1993.zbMATHGoogle Scholar
  7. [7]
    I. Ekeland and R. Temam. Convex analysis and variational problems. North-Holland Publishing Co., Amsterdam, 1976. Translated from the French, Studies in Mathematics and its Applications, Vol. 1.zbMATHGoogle Scholar
  8. [8]
    J. Hadamard. Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin, pages 49–52, 1902.Google Scholar
  9. [9]
    A. D. Ioffe and A. J. Zaslavski. Variational principles and well-posedness in optimization and calculus of variations. SIAM J. Control Optim., 38(2):566–581 (electronic), 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    R. Lucchetti. Some aspects of the connections between Hadamard and Tyhonov well-posedness of convex programs. Boll. Un. Mat. Ital. C (6), l(l):337–345, 1982.MathSciNetGoogle Scholar
  11. [11]
    R. T. Rockafellar and R. J.-B. Wets. Variational systems, an introduction. In Multifunctions and integrands (Catania, 1983), volume 1091 of Lecture Notes in Math., pages 1–54. Springer, Berlin, 1984.Google Scholar
  12. [12]
    M. A. Sychëv. Necessary and sufficient conditions in theorems of semicontinuity and convergence with a functional. Mat. Sb., 186(6):77–108, 1995.zbMATHMathSciNetGoogle Scholar
  13. [13]
    M. A. Sychëv. On the continuous dependence of the solutions of the simplest variational problems on the integrand. Siberian Math. J., 36(2):379–388, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    S. M. Ulam. A collection of mathematical problems. Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience Publishers, New York-London, 1960.zbMATHGoogle Scholar
  15. [15]
    S. Villa. Bounded hausdorff convergence of integral functionals and related well-posedness properties, preprint.Google Scholar
  16. [16]
    S. Villa. AW-convergence and well-posedness of non convex functions. J. Convex Anal., 10(2):351–364, 2003.zbMATHMathSciNetGoogle Scholar
  17. [17]
    S. Villa. Well-posedness of nonconvex integral functionals. SIAM J. Control Optim., 43(4): 1298–1312 (electronic), 2004/05.MathSciNetCrossRefGoogle Scholar
  18. [18]
    S. Villa. Well-posed problems of the calculus of variations. 2005.Google Scholar
  19. [19]
    A. Visintin. Strong convergence results related to strict convexity. Comm. Partial Differential Equations, 9(5):439–466, 1984.zbMATHMathSciNetGoogle Scholar
  20. [20]
    T. Zolezzi. Well-posed optimization problems for integral functionals. J. Optim. Theory AppL, 31(3):417–430, 1980.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    T. Zolezzi. Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal.,25(5):437–53, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    T. Zolezzi. Wellposed problems of the calculus of variations for nonconvex integrals. J. Convex Anal., 2(l–2):375–383, 1995.zbMATHMathSciNetGoogle Scholar

Copyright information

© International Federation for Information Processing 2006

Authors and Affiliations

  • S. Villa
    • 1
  1. 1.DIMAUniversità di GenovaGenovaItaly

Personalised recommendations