Eigenvalue Problems for Hemivariational Inequalities

  • D. Motreanu
  • P. D. Panagiotopoulos
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 29)


The aim of the present chapter is to formulate and study two types of eigenvalue problems for hemivariational inequalities. The first type is the most classical one, whereas the second is an eigenvalue problem for a hemivariational inequality involving a nonlinear compact operator. After giving certain existence results, we illustrate the theory with applications.


Eigenvalue Problem Compact Operator Plate Theory Real Hilbert Space Convergent Subsequence 
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  1. [1]
    B. Budiansky, Theory of Buckling and Postbuckling Behaviour of Elastic Structures. In Adv. in Appl. Mech. (ed. by Chia-Shun Yih), Acad. Press, London 1974, pp. 1–65.Google Scholar
  2. [2]
    K.C. Chang, Variational Methods for Non-Differentiable Functionals and their Applications to Partial Differential Equations. J. Math. Anal. Appl. 80 (1981), 102–129.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    P.G. Ciarlet and P. Rabier, Les equations de von Kármán. Lect. Notes in Math. Vol. 826, Springer Verlag, Berlin 1980.MATHGoogle Scholar
  4. [4]
    F.H. Clarke, Optimization and Nonsmooth Analysis, J. Wiley and Sons, New York, 1983.MATHGoogle Scholar
  5. [5]
    J. Ekeland, On the Variational Principle, J.Math. Anal. Appl. 47 (1974), 324–353.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    G.Fichera, Existence Theorems in Elasticity. In: Encyclopedia of Physics (ed. by S. Flüge) Vol VI a/2. Springer-Verlag, Berlin 1972.Google Scholar
  7. [7]
    I. Karamanlis, Buckling Problems in Composite von Kärmän Plates. Doct. Thesis, Aristotle University Dept. of Civil Eng. 1991.Google Scholar
  8. [8]
    H.N. Karamanlis, P.D. Panagiotopoulos, The Eigenvalue Problem in Hemivariational Inequalities and its Application to Composite Plates. Journal of the Mech. Behaviour of Materials (Freund Publ. House, Tel Aviv) 5, No. 1, (1993), 67–76.Google Scholar
  9. [9]
    C.Lefter and D.Motreanu, Critical Point Methods in Nonlinear Eigenvalue Problems with Discontinuities, in: International Series of Numerical Mathematics, Vol. 107 Birkhäuser Verlag, Basel, Boston 1992.Google Scholar
  10. [10]
    D.Motreanu, Existence for Minimization with Nonconvex Constraints, J. Math. Anal. Appl. 117 (1986), 128–137.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    D.Motreanu and P.D. Panagiotopoulos, Hysteresis: The Eigenvalue Problem for Hemivariational Inequalities, in “Models of Hysteresis” (ed. by A. Visintin), Pitman Research Notes in Math. 286, Longman Harlow, 1993, pp. 102–117.Google Scholar
  12. [12]
    Z. Naniewicz and P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, N.York, 1995.Google Scholar
  13. [13]
    J. Naumann, H.V. Wenk, On Eigenvalue Problems for Variational Inequalites, Rendiconti di Matem. (3) Vol. 9 Serie VI(1976), 439–463.Google Scholar
  14. [14]
    P.D. Panagiotopoulos, Hemivariational Inequalities. Application to Mechanics and Engineering, Springer Verlag, N.York, Berlin, 1993. Google Scholar
  15. [15]
    P.D. Panagiotopoulos, Semicoercive Hemivariational inequalities. On the delami- nation of composite plates. Quart, of Appl. Math. XLVII (1989), 611–629.MathSciNetGoogle Scholar
  16. [16]
    P.D. Panagiotopoulos and G. Stavroulakis, The Delamination Effect in Laminated von Kármán Plates under Unilateral Boundary Conditions. A Variational- Hemivariational Inequality Approach, Journal of Elasticity 23 (1990), 69–96.MathSciNetMATHGoogle Scholar
  17. [17]
    P.D. Panagiotopoulos,Inequaiity Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Verlag, Boston, Basel, 1985 (Russian Translation MIR Publ. Moscow 1989).Google Scholar
  18. [18]
    P.D. Panagiotopoulos, Coercive and Semicoercive Hemivariational Inequalities. Nonlinear Anal. T.M.A. 16 (1991), 209–231.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    P.D. Panagiotopoulos, Hemivariational Inequalites and Substationarity in the Static Theory of von Kärmän Plates. ZAMM 65 (1985), 219–229.MathSciNetCrossRefGoogle Scholar
  20. [20]
    P.D. Panagiotopoulos and G. Stavroulakis, A Variational-Hemivariational Inequality Approach to the Laminated Plate Theory under Subdifferential Boundary Conditions. Quart, of Appl. Math. XLVI (1988), 409–430.MathSciNetGoogle Scholar
  21. [21]
    P.D. Panagiotopoulos, Hemivariational Inequalities and their Applications. In: Topics in Nonsmooth Mechanics (ed. J.J. Moreau, P.D. Panagiotopoulos and J. Strang), Birkhäuser Verlag, Boston, Basel, 1988.Google Scholar
  22. [22]
    P.D. Panagiotopoulos, Non-Convex Superpotentials in the Sense of F.H. Clarke and Applications. Mech. Res. Comm. 8 (1981), 335–340.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    P.D. Panagiotopoulos, Nonconvex Energy Functions. Hemivariational Inequalities and Substationarity Principles. Acta Mechanica 42 (1983), 160–183.Google Scholar
  24. [24]
    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
  25. [25]
    P.H. Rabinowitz, Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations, Nonlinear Analysis: A collection of papers in honor of E. Rӧthe, Academic Press, New-York, 1978, pp. 161–177.Google Scholar
  26. [26]
    R.T. Rockafellar, La Théorie des Sous-Gradients et ses Applications á L’Optimization. Fonctions Convexes et Non-convexes, Les Presses de 1’ Université de Montréal, Montréal 1979.Google Scholar
  27. [27]
    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.MathSciNetMATHGoogle Scholar
  28. [28]
    S. Timoshenko and J. Gere, Theory of Elastic Stability (2nd Edition) McGraw Hill, N. York, 1961.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • D. Motreanu
    • 1
  • P. D. Panagiotopoulos
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of IasiRomania
  2. 2.Department of Civil EngineeringAristotle UniversityThessalonikiGreece
  3. 3.Faculty of Mathematics and PhysicsRWTH AachenGermany

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