Mathematical Preliminaries

  • Jaroslav Haslinger
  • Markku Miettinen
  • Panagiotis D. Panagiotopoulos
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 35)

Abstract

In order to keep the presentation as self-contained we decided to start with basic results, which will be needed in subsequent parts of the monograph. First, we give a survey of function spaces of Sobolev and Bochner type and their properties, then we mention elements of convex and nonconvex analysis. We recall also main results on the approximation of monotone problems of elliptic and parabolic type. Subsequent parts of this monograph will present an extension of these results to a larger class of problems, involving nonmonotone and nondifferentiable inclusions.

Keywords

Finite Element Method Variational Inequality Convex Subset Bilinear Form Reflexive Banach Space 
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. Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York.MATHGoogle Scholar
  2. Aubin, J.-P. and Clarke, F. H. (1979). Shadow prices and duality for a class of optimal control problems. SIAM J. Control Optimization, 17: 567–586.MathSciNetMATHCrossRefGoogle Scholar
  3. Aubin, J.-P. and Ekeland, I. (1984). Applied Nonlinear Analysis. J. Wiley and Sons, New York.MATHGoogle Scholar
  4. Aubin, J.-P. and Frankowska, H. (1990). Set-valued analysis, volume 2 of Systems & Control: Foundations & Applications. Birkhäuser, Boston.Google Scholar
  5. Axelsson, O. and Barker, V. A. (1984). Finite Element Solution of Boundary Value Problems. Academic Press, Orlando.MATHGoogle Scholar
  6. Barbu, V. (1993). Analysis and control of nonlinear infinite-dimensional systems, volume 190 of Mathematics in Science and Engineering. Academic Press, Boston.Google Scholar
  7. Brézis, H. (1973). Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert. North-Holland Publ. Co. Amsterdam and American Elsevier Publ. Co., New York.Google Scholar
  8. Browder, F. E. and Hess, P. (1972). Nonlinear mappings of monotone type in Banach spaces. J. Funct. Anal., 11: 251–294.MathSciNetMATHCrossRefGoogle Scholar
  9. Ciarlet, P. G. (1978). The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, New York, Oxford.Google Scholar
  10. Clarke, F. (1983). Optimization and Nonsmooth Analysis. J. Wiley, New York.Google Scholar
  11. Doktor, P. (1973). On the density of smooth functions in certain subspaces of Sobolev spaces. Comment. Math. Univ. Carolin., 14: 609–622.MathSciNetMATHGoogle Scholar
  12. Duvaut, G. and Lions, J. L. (1976). Inequalities in Mechanics and Physics. Springer-Verlag, Berlin, Heidelberg, New York.CrossRefGoogle Scholar
  13. Ekeland, I. and Temam, R. (1976). Convex Analysis and Variational Problems. North-Holland, Amsterdam.MATHGoogle Scholar
  14. Fučik, S. and Kufner, A. (1980). Nonlinear Differential Equations. Studies in Applied Mechanics 2. Elsevier, Amsterdam, New York.Google Scholar
  15. Glowinski, R. (1984). Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, New York.MATHGoogle Scholar
  16. Glowinski, R., Lions, J. L., and Trémoliéres, R. (1981). Numerical analysis of variational inequalities, volume 8 of Studies in Mathematics and its Applications. North Holland, Amsterdam, New York.Google Scholar
  17. Haslinger, J., Hlavâcek, I., and Necas, J. (1996). Numerical methods for unilateral problems in solid mechanics. In Ciarlet, P. G. and Lions, J. L., editors, Handbook of Numerical Analysis. North Holland.Google Scholar
  18. Haslinger, J. and Neittaanmäki, P. (1996). Finite Element Approximation for Optimal Shape, Material and Topology Design. J. Wiley, second edition.Google Scholar
  19. Hlavâcek, I., Haslinger, J., Necas, J., and Lovisek, J. (1988). Numerical Solution of Variational Inequalities. Springer Series in Applied Mathematical Sciences 66. Springer-Verlag, New York.CrossRefGoogle Scholar
  20. Kikuchi, N. and Oden, J. T. (1988). Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM Studies in Applied Mathematics 8. SIAM, Philadelphia.CrossRefGoogle Scholar
  21. Kufner, A., John, O., and Flicik, S. (1977). Function Spaces. Noordhoff International Publishing Leyden; Academia, Prague,.Google Scholar
  22. Křižek, M. and Neittaanmäki, P (1990). Finite Element Approximation of Variational Problems and Applications. Longman Scientific & Technical, Harlow.Google Scholar
  23. Landes, R. and Mustonen, V. (1987). A strongly nonlinear parabolic intial value problem. Ark. f. Mat., 25: 29–40.MathSciNetMATHCrossRefGoogle Scholar
  24. Lions, J. L. (1969). Quelques Méthodes de résolution des problèmes aux limites non linéaires. Dunod/Gauthier-Villairs, Paris.MATHGoogle Scholar
  25. Miettinen, M. (1996). A parabolic hemivariational inequality. Nonlinear Analysis, 26: 725–734.MathSciNetMATHCrossRefGoogle Scholar
  26. Moreau, J. J. (1967). Fonctionnelles Convexes. Séminaire sur les équations aux dérivées partielles. Collège de France. Paris.Google Scholar
  27. Neittaanmäki, P. and Tiba, D. (1994). Optimal control of nonlinear parabolic systems. Theory, algorithms, and applications., volume 179 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York.Google Scholar
  28. Nečas, J. (1967). Les Méthodes Directes en Théorie des Equations Elliptiques. Masson, Paris.MATHGoogle Scholar
  29. Nečas, J. and Hlavâcek, I. (1981). Mathematical Theory of Elastic and ElastoPlastic Bodies: An Introduction. Elsevier, Amsterdam.Google Scholar
  30. Rockafellar, R. T. (1969). Convex Analysis. Princeton Univ. Press, Princeton.Google Scholar
  31. Thomée, V. (1984). Galerkin finite element methods for parabolic problems. Springer-Verlag, Berlin, Heidelberg, New York.Google Scholar
  32. Trémoliéres, R. (1972). Inéquations variationnelles: existence, approximations, résolution. PhD thesis, Université de Paris V I.Google Scholar
  33. Yosida, K. (1965). Functional Analysis. Springer-Verlag, Berlin.MATHGoogle Scholar
  34. Zeidler, E. (1990a). Nonlinear functional analysis and its applications. II/A Linear monotone operators. Springer-Verlag, Berlin, New York.MATHCrossRefGoogle Scholar
  35. Zeidler, E. (1990b). Nonlinear functional analysis and its applications. II/B Nonlinear monotone operators. Springer-Verlag, Berlin, New York.MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Jaroslav Haslinger
    • 1
  • Markku Miettinen
    • 2
  • Panagiotis D. Panagiotopoulos
    • 3
  1. 1.Charles UniversityCzech Republic
  2. 2.University of JyväskyläFinland
  3. 3.Aristotle UniversityGreece

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