Minimax Theorems for Locally Lipschitz Functionals and Applications

  • Maria do Rosário Grossinho
  • Stepan Agop Tersian
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 52)


The purpose of this chapter is to present several variants of minimization and mountain-pass theorems for nondifferentiable functionals. We assume that the given functionals are locally Lipschitz so that their generalized gradients can be defined (cf. Clarke [Cl] ). A general critical point theory for locally Lipschitz functionals was developed by K. C. Chang [Ch1], extending the concept of a critical point, the Palais—Smale condition and the deformation lemma. Critical point results have also been obtained in other nondifferentiable settings. We refer to Degiovanni & Marzocchi [DM], Corvellec, Degiovanni & Marzocchi [CDM] for the case of continuous functionals and to Ribarska, Tsachev & Krastanov [RTK1], Ioffe & Schwartzman [IS] for the case of discontinuous functionals.


Generalize Gradient Directional Derivative Convergent Subsequence Critical Point Theory Minimax Theorem 
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  1. [AG1]
    Adly S, Goeleven D. Periodic solutions for a class of hemivariational inequalities. Comm. Appl. Nonl. Anal., 1995; 2:45–57.MathSciNetzbMATHGoogle Scholar
  2. [AG2]
    Adly S, Goeleven D. Homoclinic orbits for a class of hemivariational inequalities. Applicable Analysis, 1995; 58:229–240.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Au]
    Aubin, Jean-Pierre. L ’analyse Non Linéaire et ses Motivations Economique. Paris: Masson, 1984.Google Scholar
  4. [AEk]
    Aubin, Jean-Pierre and Ekeland, Ivar. Applied Nonlinear Analysis. N.Y.: John Wiley amp; Sons, 1984.zbMATHGoogle Scholar
  5. [BN]
    Brezis H, Nirenberg L. Remarks on finding critical points. Comm. Pure Appl. Math., 1991;XLIV:939–963.MathSciNetCrossRefGoogle Scholar
  6. [CLW]
    Caklovic L, Li S, Willem M. A note on Palais-Smale condition and coercivity. Diff. Int. Eq., 1990;3:799–800MathSciNetzbMATHGoogle Scholar
  7. [Cha]
    Chabrowski, Jan. Variational Methods for Potential Equations: with Applications to Nonlinear Elliptic Equations. Berlin: de Gruyter, 1997.CrossRefzbMATHGoogle Scholar
  8. [Ch1]
    Chang KC. Variational methods for non-differentiable functionals and their applications to PDE. J. Math. Anal. Appl., 1981;80:102–129.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Cl]
    Clarke, Frank. Optimization and Nonsmooth Analysis. N.Y.: John Wiley & Sons, 1983.zbMATHGoogle Scholar
  10. [CMG]
    Corvellec JN, Degiovanni M, Marzocchi M. Deformation properties for continuous functionals and critical point theory. Topol. Methods Nonlin. Anal. 1993;1:151-.MathSciNetzbMATHGoogle Scholar
  11. [CG]
    Costa DG, Gonçalves JVA. Critical point theory for nondifferentiable functionals and applications. J. Math. Anal. Appl., 1990;153:470–485.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [DM]
    Degiovanni M., Marzocchi M., A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl., 1994;167:73–100.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [Ek2]
    Ekeland, Ivar. Convexiy Methods in Hamiltoniam Mechanics. N.Y.: Springer-Verlag, 1990.CrossRefGoogle Scholar
  14. [GM1]
    Ghoussoub N, Preiss D. A general mountain pass principle for locating and classifying critical points. Ann. Inst. Henri Poincaré, 1989;6,5:321–330.MathSciNetzbMATHGoogle Scholar
  15. [GT1]
    Grossinho MR, Tersian S. Critical point theory for locally Lipschitz functionals and applications to fourth order problems. Proceedings of XXVIII-th Spring Conference of U.B.M., Sofia, 1999:99–106.Google Scholar
  16. [IS]
    Ioffe A., Schwartzman E., Metric critical point theory, 1. Morse regularity and homotopic stability of a minimum. J. Math. Pures Appl., 1996;75:125–153.MathSciNetzbMATHGoogle Scholar
  17. [KRT1]
    Ribarska N., Tsachev T. and Krastanov M. The intrinsic mountain pass principle. C. R. Acad. Sci. Paris, Ser I, 1999;329:350–358.MathSciNetGoogle Scholar
  18. [KRT2]
    Ribarska N., Tsachev T. and Krastanov M. Deformation Lemma, Ljusternik-Schnirelmann Theory and Mountain Pass theorem on C1-Finsler Manifolds. Serdica Math. J., 1991;21:239–266.MathSciNetGoogle Scholar
  19. [KRT3]
    Ribarska N., Tsachev T. and Krastanov M. On the general mountain pass principle of Ghoussoub-Preiss. Mathematica Balkanika (N.S.), 1991;5:399–404.MathSciNetGoogle Scholar
  20. [Panl]
    Panagiotopoulos, Panagiotis. Hemivariational Inequalities. Applications in Mechanics and Engineering. Berlin: Springer-Verlag, 1993.CrossRefzbMATHGoogle Scholar
  21. [Pan2]
    Panagiotopoulos PD. Nonconvex energy function, hemivariational inequalities and substationary principles. Acta Mech. 1983;48:160–183.MathSciNetCrossRefGoogle Scholar
  22. [Pan3]
    Panagiotopoulos PD. Coercive and semicoercive hemivariational inequalities. Nonlinear Anal., 1991;16:209–231.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [Wil1]
    Willem M. Lecture notes on critical point theory, Fundação Universidade de Brasília, 199, 1983.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Maria do Rosário Grossinho
    • 1
    • 2
  • Stepan Agop Tersian
    • 3
  1. 1.ISEGUniversidade Técnica de LisboaPortugal
  2. 2.CMAFUniversidade de LisboaPortugal
  3. 3.University of RousseBulgaria

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