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Critical Points for Nonsmooth Functionals

  • D. Motreanu
  • V. Rădulescu
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 67)

Abstract

The present Chapter deals with critical point theory for three different nonsmooth situations. First, we set forth the critical point theory for locally Lipschitz functionals, following the approach of Chang [4] and Clarke [5]. Then a critical point theory is described for nonsmooth functionals expressed as a sum of a locally Lipschitz function and a convex, proper and lower semicontinuous function, using the development in Motreanu and Panagiotopoulos [26]. Finally, the critical point theory for continuous functionals defined on a complete metric space as introduced by Degiovanni and Marzocchi [7] is presented.

Keywords

Variational Method Critical Point Theory Lower Semicontinuous Function Hemivariational Inequality Nonempty Closed Subset 
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]
    S. Adly and G. Buttzaao and M. Thera,Critcal poins for nonsmooth energy function and applicatoins, Nonlinear. Anal. 32 (1998), 711–718.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    H. Brézis, J.-M. Coron and L. Nirenberg, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Commun. Pure Appl. Math. 33 (1980), 667–684.MATHCrossRefGoogle Scholar
  4. [4]
    K.-C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    F. H. Clarke, Optimization and Nonsmooth Analysis, New York, John WileyInterscience, 1983.MATHGoogle Scholar
  6. [6]
    M. Degiovanni, Nonsmooth critical point theory and applications, Second World Congress of Nonlinear Analysts (Athens, 1996), Nonlinear Anal. 30 (1997), 8999.MathSciNetCrossRefGoogle Scholar
  7. [7]
    M. Degiovanni and M. Marzocchi, A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl., IV. Ser. 167 (1994), 73–100.MathSciNetMATHGoogle Scholar
  8. [8]
    G. Dinch, P. Jebelean and D. Motreanu, Existence and approximation for a general class of differential inclusions, Houston J. Math. 28 (2002), 193–215.MathSciNetGoogle Scholar
  9. [9]
    I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324353.Google Scholar
  10. [10]
    I. Ekeland, Nonconvex minimization problems, Bull. (New Series) Amer. Math. 1 (1979), 443–474.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    M. Fundo, Hemivariational inequalities in subspaces of Lr(Q)(p > 3), Nonlinear Anal. 33 (1998), 331–340.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    L. Gasinski and N. S. Papageorgiou, Solutions and multiple solutions for quasi-linear hemivariational inequalities at resonance, Proc. R. Soc. Edinb. 131 (2001), 1091–1111.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6 (1989), 321–330.MathSciNetMATHGoogle Scholar
  14. [14]
    D. Goeleven and D. Motreanu, Minimax methods of Szulkin’s type unilateral problems, Functional analysis. Selected topics. Dedicated to the memory of the late Professor P. K. Kamthan (Jain, Pawan K. (ed.)), Narosa Publishing House, New Delhi, 158–172, 1998.Google Scholar
  15. [15]
    D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities, Theory, Methods and Applications,Volume I: Unilateral Analysis and Unilateral Mechanics, Kluwer Academic Publishers, Dordrecht/Boston/London, to appear.Google Scholar
  16. [16]
    D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities, Theory, Methods and Applications,Volume II: Unilateral Problems, Kluwer Academic Publishers, Dordrecht/Boston/London, to appear.Google Scholar
  17. [17]
    J. Haslinger and D. Motreanu, Hemivariational inequalities with a general growth condition: existence and approximation, submitted.Google Scholar
  18. [18]
    L. T. Hu, Homotopy Theory, Academic Press, New York, 1959.MATHGoogle Scholar
  19. [19]
    N. C. Kourogenis and N. S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, Kodai Math. J. 23 (2000), 108–135.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    S. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian, J. Differ. Equations 182 (2002), 108–120.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.Google Scholar
  22. [22]
    D. Motreanu, Existence and critical points in a general setting, Set-Valued Anal. 3 (1995), 295–305.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    D. Motreanu, Eigenvalue problems for variational-hemivariational inequalities in the sense of P. D. Panagiotopoulos, Nonlinear Anal. 47 (2001), 5101–5112.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    D. Motreanu and Z. Naniewicz, Discontinuous semilinear problems in vector-valid function spaces, Differ. Integral Equ. 9 (1996), 581–598.MathSciNetMATHGoogle Scholar
  25. [25]
    D. Motreanu and Z. Naniewicz, A topological approach to hemivariational inequalities with unilateral growth condition, J. Appl. Anal. 7 (2001), 23–41.MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optimization and its Applications 29, Kluwer Academic Publishers, Dordrecht/Boston/London, 1998.Google Scholar
  27. [27]
    D. Motreanu and Cs. Varga, Some critical points results for locally Lipschitz functionals, Commun. Appl. Nonlinear Anal. 4 (1997), 17–33.MathSciNetMATHGoogle Scholar
  28. [28]
    D. Motreanu and Cs. Varga, A nonsmooth equivariant minimax principle, Commun. Appl. Anal. 3 (1999), 115–130.MathSciNetMATHGoogle Scholar
  29. [29]
    J. R. Munkres, Elementary Differential Topology, Princeton University Press, Princeton, 1966.Google Scholar
  30. [30]
    Z. Naniewicz and P. D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications, Pure and Applied Mathematics, Marcel Dekker 188, New York, 1994.Google Scholar
  31. [31]
    R. S. Palais and S. Smale, A generalized Morse theory, Bull. Am. Math. Soc. 70, (1964) 165–172.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    J.-P. Penot, Well-behaviour, well-posedness and nonsmooth analysis, PLISKA, Stud. Math. Bulg. 12 (1998), 141–190.MathSciNetMATHGoogle Scholar
  33. [33]
    P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. in Math. 65, Amer. Math. Soc., Providence, 1986.Google Scholar
  34. [34]
    V. Ràdulescu, Mountain pass theorems for non-differentiable functions and applications, Proc. Japan Acad., Ser. A 69 (1993), 193–198.MATHGoogle Scholar
  35. [35]
    N. K. Ribarska, Ts. Y. Tsachev, M. I. Krastanov, Speculating about mountains, Serdica Math. J. 22 (1996), 341–358.MathSciNetMATHGoogle Scholar
  36. [36]
    B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems. Nonlinear operator theory, Math. Comput. Modelling 32 (2000), 1485–1494.MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    B. Ricceri, New results on local minima and their applications, in: Equilibrium Problems and Variational Models (editors: F. Giannessi, A. Maugeri and P. M. Pardalos), Taormina, 2–5 December 1998, Kluwer Academic Publishers, Dordrecht/Boston/London, 2001, pp. 255–268.Google Scholar
  38. [38]
    W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics, New York, 1973.MATHGoogle Scholar
  39. [39]
    E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.MATHGoogle Scholar
  40. [40]
    A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. Henri Poincaré. Anal. Non Linéaire 3 (1986), 77–109.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • D. Motreanu
    • 1
  • V. Rădulescu
    • 2
  1. 1.Department of MathematicsUniversity of PerpignanPerpignanFrance
  2. 2.Department of MathematicsUniversity of CraiovaCraiovaRomania

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