Advertisement

Semilinear Hemivariational Inequalities with Dirichlet Boundary Condition

  • Dumitru Motreanu
  • Zdzisław Naniewicz
Chapter
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 1)

Abstract

The paper studies nonsmooth semilinear elliptic boundary value problems which are expressed in the form of hemivariational inequalities. The approach relies on nonsmooth variational methods using essentially a general unilateral growth condition and a new concept of solution. The known results are recovered without additional assumptions.

Keywords

Boundary value problems hemivariational inequalities nonsmooth analysis nonconvex potential. 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, New York, 1984.zbMATHGoogle Scholar
  3. [3]
    K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    E H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983.zbMATHGoogle Scholar
  5. [5]
    G. Dinca, P. D. Panagiotopoulos, and G. Pop, Inéqualités hémivariationnelles semi-coercives sur des ensembles convexes, C. R. Acad. Sci. Paris 320, Serie I (1995), 1183–1186.Google Scholar
  6. [6]
    G. Duvaut and J. L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972.zbMATHGoogle Scholar
  7. [7]
    I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, Vol. 28, SIAM, Philadelphia, 1999.CrossRefGoogle Scholar
  8. [8]
    D. Y. Gao, Duality Principles in Nonconvex Systems, Nonconvex Optimization and Its Applications, Vol. 39, Kluwer Academic Publishers, Dordrecht/Boston/London, 2000.Google Scholar
  9. [9]
    D. Goeleven and D. Motreanu, Eigenvalue and dynamic problems for variational and hemivariational inequalities, Comm. Appl. Nonl. Analysis 3 (1996), 1–21.MathSciNetzbMATHGoogle Scholar
  10. [10]
    D. Goeleven and D. Motreanu, Minimax methods of Szulkin’s type in unilateral problems, in: Functional Analysis–Selected Topics (Ed. P. K. Jain ), Narosa Publishing House, New Delhi, India, 1998, p. 169–183.Google Scholar
  11. [11]
    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
  12. [12]
    J. Haslinger, M. Miettinen, and P. D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1999.CrossRefGoogle Scholar
  13. [13]
    L. I. Hedberg, Two approximation problems in function spaces, Ark. Mat. 16 (1978), 51–81.MathSciNetzbMATHGoogle Scholar
  14. [14]
    E. S. Mistakidis and G. E. Stavroulakis, Nonconvex Optimization in Mechanics. Smooth and Nonsmooth Algoritms, Heuristic and Engineering Applications by the F.E.M., Kluwer Academic Publishers, Dordrecht, Boston, London, 1998.Google Scholar
  15. [15]
    D. Motreanu, Existence of critical points in a general setting, Set-Valued Anal. 3 (1995), 295–305.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    D. Motreanu and Z. Naniewicz, Discontinuous semilinear problems in vector-valued function spaces, Differ. Int. Equations 9 (1996), 581–598.MathSciNetzbMATHGoogle Scholar
  17. [17]
    D. Motreanu and Z. Naniewicz, A topological approach to hemivariational inequalities with unilateral growth conditions, J. Appl. Anal. 7 (2001), 2341.Google Scholar
  18. [18]
    D. Motreanu and P. D. Panagiotopoulos, An eigenvalue problem for a hemivariational inequality involving a nonlinear compact operator, Set-Valued Anal. 3 (1995), 157–166.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    D. Motreanu and P. D. Panagiotopoulos, Nonconvex energy functions, Related eigenvalue hemivariational inequalities on the sphere and applications, J. Global Optimiz. 6 (1995), 163–177.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    D. Motreanu and P. D. Panagiotopoulos, On the eigenvalue problem for hemivariational inequalities: existence and multiplicity of solutions, J. Math. Anal. Appl. 197 (1996), 75–89.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    D. Motreanu and P. D. Panagiotopoulos, Double eigenvalue problems for hemivariational inequalities, Arch. Rational Mech. Anal. 140 (1997), 225–251.MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. [22]
    D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer Academic Publishers, Dordrecht, Boston, London, 1999.Google Scholar
  23. [23]
    Z. Naniewicz, Hemi-variational inequalities: Static problems, Encyclopedia of Optimization, Vol. II, p. 389–409, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.Google Scholar
  24. [24]
    Z. Naniewicz, Semicoercive variational-hemivariational inequalities with unilateral growth condition, J. Global Optim. 17 (2000), 317–337.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Z. Naniewicz, Hemivariational inequalities with functions fulfilling directional growth condition, Appl. Anal. 55 (1994), 259–285.MathSciNetzbMATHGoogle Scholar
  26. [26]
    Z. Naniewicz, Hemivariational inequalities as necessary conditions for optimality for a class of nonsmooth nonconvexfunctionals, Nonlinear World 4 (1997), 117–133.MathSciNetzbMATHGoogle Scholar
  27. [27]
    Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, New York, 1995.Google Scholar
  28. [28]
    P. D. Panagiotopoulos, Non-convex superpotentials in the sense of F.H. Clarke and applications, Mech. Res. Comm 8 (1981), 335–340.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    P. D. Panagiotopoulos, Non-convex energy functionals. Aplications to non-convex elastoplasticity, Mech. Res. Comm 9 (1982), 23–29.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Verlag, Basel, 1985.zbMATHCrossRefGoogle Scholar
  31. [31]
    P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, New York, 1993.CrossRefGoogle Scholar
  32. [32]
    G. Pop, P. D. Panagiotopoulos and Z. Naniewicz, Variationalhemivariational inequalities for multidimensional superpotential laws, Nu-mer. Funct. Anal. Optim. 18 (1997), 827–856.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Sec. in Math. No. 65, Amer. Math Soc., Providence, R. I., 1986.Google Scholar
  34. [34]
    A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. Henri Poincaré, Analyse Non Linéaire 3 (1986), 77–109.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Dumitru Motreanu
    • 1
  • Zdzisław Naniewicz
    • 2
  1. 1.Department of MathematicsUniversity of PerpignanPerpignanFrance
  2. 2.Faculty of Mathematics and ScienceCardinal Stefan Wyszyński UniversityWarsawPoland

Personalised recommendations

Cite chapter

Buy options