Hemivariational Inequalities. Existence and Approximation Results

  • P. D. Panagiotopoulos
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 288)


The theory of variational inequalities is closely connected to the notion of superpotential introduced by Moreau1 for convex generally non-differentiable and nonfinite energy functionals. If Φ is such a functional, a superpotential relation (material law or boundary condition) in the sense of Moreau has the form
$$f \in \partial \Phi \left( u \right)$$
where f and u are a “force” and a “flux” respectively in the terminology of Onsager’s theory and ∂ denotes the subdifferential2, which is a monotone multivalued operator. Such a law allows the derivation of variational inequalities3, 4 which are expressions of the principle of virtual or complementary virtual work (or power) in static problems and of d’ Alembert’s principle in dynamic problems.


Variational Inequality Heat Conduction Problem Hemivariational Inequality Hausdorff Topological Vector Space Multivalued Operator 
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  1. 1.
    Moreau, J.J., La notion de sur-potentiel et les liaisons unilatérales en élastostatique, C.R.Acad.Sc.Paris, 267, 954, 1968.MATHMathSciNetGoogle Scholar
  2. 2.
    Rockafellar, R.T., Convex Analysis, Princeton Univers. Press, Princeton, 1970.Google Scholar
  3. 3.
    Duvaut, G. and Lions, J.L., Les inéquations en Mécanique et en Physique, Dunod, Paris, 1972, chap. 1, 2.Google Scholar
  4. 4.
    Fichera, G., Boundary Value Problems in Elasticity with Unilateral Constraints, in Encyclopedia of Physics VIa/2, S.Flügge Ed., Springer-Verlag, Berlin, Heidelberg, New York, 1972.Google Scholar
  5. 5.
    Panagiotopoulos, P.D., Nonconvex Superpotentials in the sense of F.H. Clarke and Applications, Mech.Res.Comm., 8, 335, 1981.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Panagiotopoulos, P.D., Nonconvex Energy Functionals. Application to Nonconvex Elastoplasticity, Mech.Res.Comm., 9, 23, 1982.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Panagiotopoulos, P.D., Non-convex Energy Functions. Hemivariational Inequalities and Substationarity Principles, Acta Mechanica, 48, 111, 1983.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Panagiotopoulos, P.D., Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions,Birkhäuser Verlag, Basel, Boston, 1984 (in press).Google Scholar
  9. 9.
    Clarke, F.H., Generalized gradients and applications, Trans.A.M.S., 205, 247, 1975.CrossRefMATHGoogle Scholar
  10. 10.
    Clarke, F.H., Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.MATHGoogle Scholar
  11. 11.
    Rockafellar, R.T., La théorie des sous-gradients et ses applications d l’optimisation. Fonctions convexes et non-convexes, Les presses de l’Université de Montréal, Montréal, 1979.Google Scholar
  12. 12.
    Rauch, J., Discontinuous Semilinear Differential Equations and Multiple Valued Maps, Proc.A.M.S., 64, 277, 1977.CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    McKenna, P.J. and Rauch, J., Strongly Nonlinear Perturbations of Nonnegative Boundary Value Problems with Kernel, J.Diff.Eq.,28, 253, 1978.Google Scholar
  14. 14.
    Castaing, C. and Valadier, M., Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin, 1977, 65.CrossRefMATHGoogle Scholar
  15. 15.
    Chang, K.C., Variational Methods for Nondifferentiable Functionals and their Applications to Partial Differential Equations, J.Math. Anal.Appl., 80, 102, 1981.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Lions, J.L., Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod-Gauthier Villars, Paris, 1969, 53.Google Scholar
  17. 17.
    Ekeland, I. and Temam, R., Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976, 239.MATHGoogle Scholar
  18. 18.
    Landesman, E.M. and Lazer, A.C., Nonlinear Perturbations of Linear Elliptic Boundary Value problems at Resonance, J.of Math. and Mech., 19, 609, 1970.MATHMathSciNetGoogle Scholar
  19. 19.
    Rockafellar, R.T., Generalized Directional Derivatives and Sub-gradients of Nonconvex Functions, Can.J.Math., XXXII, 257, 1980.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Panagiotopoulos, P.D., Nonconvex Problems of Semipermeable Media and Related Topics. To appear in ZAMM, 64, 1984.Google Scholar

Copyright information

© Springer-Verlag Wien 1985

Authors and Affiliations

  • P. D. Panagiotopoulos
    • 1
  1. 1.School of TechnologyAristotelian UniversityThessalonikiGreece

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