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Archive for Rational Mechanics and Analysis

, Volume 100, Issue 2, pp 149–189 | Cite as

General existence theorems for unilateral problems in continuum mechanics

  • Claudio Baiocchi
  • Giuseppe Buttazzo
  • Fabio Gastaldi
  • Franco Tomarelli
Article

Abstract

The problem of minimizing a possibly non-convex and non-coercive functional is studied. Either necessary or sufficient conditions for the existence of solutions are given, involving a generalized recession functional, whose properties are discussed thoroughly. The abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region.

Keywords

Neural Network Complex System Nonlinear Dynamics Electromagnetism Applied Force 
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. [AF]
    Acerbi, E., & Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984), 125–145.Google Scholar
  2. [A]
    Anzellotti, G., A class of convex non-coercive functionals and masonry-like materials. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 261–307.Google Scholar
  3. [ABD]
    Anzellotti, G., Buttazzo, G., & Dal Maso, G., Dirichlet problems for demicoercive functionals. Nonlinear Anal. 10 (1986), 603–613.Google Scholar
  4. [B]
    Baiocchi, C., Disequazioni variazionali non coercive. Proc. Convegno Intern. M. Picone-L. Tonelli, Roma (1985) (to appear).Google Scholar
  5. [BC]
    Baiocchi, C., & Capelo, A., Variational and quasivariational inequalities: applications to free boundary problems. J. Wiley and Sons, Chichester (1984).Google Scholar
  6. [BGT1]
    Baiocchi, C., Gastaldi, F., & Tomarelli, F., Inéquations variationnelles non coercives. C. R. Acad. Sci. Paris. Ser. I Math. 299 (1984), 647–650.Google Scholar
  7. [BGT2]
    Baiocchi, C., Gastaldi, F., & Tomarelli, F., Some existence results on non-coercive variational inequalities. Ann. Scuola Norm. Sup. Pisa cl. Sci IV, 13 (1986), 617–659.Google Scholar
  8. [Ba]
    Ball, J., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–406.Google Scholar
  9. [BGK]
    Boieri, P., Gastaldi, F., & Kinderlehrer, D., Existence, uniqueness and regularity results for the two bodies contact problem. Appl. Math. Optim. 15 (1987), 251–277.Google Scholar
  10. [Bou]
    Bourbaki, N., Eléments de Mathématique—Espaces Vectoriels Topologiques, Ch. 1 et 2. Act. Sci. Ind., 1189, Hermann, Paris (1966).Google Scholar
  11. [BES]
    Busemann, H., Ewald, G., & Shepard, G. C., Convex bodies and convexity on Grassmann cones. Parts I–IV, Math. Ann. 151 (1963), 1–41.Google Scholar
  12. [Bu]
    Buttazzo, G., Su una definizione generate dei Γ-limiti. Boll. Un. Mat. Ital. (5) 14-B (1977), 722–744.Google Scholar
  13. [C]
    Ciarlet, P. G., Elasticité tridimensionnelle. Masson, Paris (1986).Google Scholar
  14. [CG]
    Ciarlet, P. G., & Geymonat, G., Sur les lois de comportement en élasticité nonlinéaire compressible. C. R. Acad. Sci. Paris. Ser. I Math. 295 (1982), 423–426.Google Scholar
  15. [CN1]
    Ciarlet, P. G., & Nečas, J., Unilateral problems in nonlinear, three dimensional elasticity. Arch. Rational Mech. Anal. 87 (1985), 319–338.Google Scholar
  16. [CN2]
    Ciarlet, P. G., & Nečas, J., Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987), 171–188.Google Scholar
  17. [DaL]
    Dal Maso, G., & Longo, P., Γ-limits of obstacles. Ann. Mat. Pura Appl. 128 (1980), 1–50.Google Scholar
  18. [D]
    De Giorgi, E., Γ-convergenza e G-convergenza. Boll. Un. Mat. Ital. (5) 14-A (1977), 213–224.Google Scholar
  19. [DF]
    De Giorgi, E., & Franzoni, T., Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 58 (1975), 842–850.Google Scholar
  20. [DL]
    Duvaut, G., & Lions, J. L., Inequalities in mechanics and physics. Springer-Verlag, Berlin Heidelberg New York (1976).Google Scholar
  21. [FZ]
    Federer, H., & Ziemer, W., The Lebesgue set of a function whose distribution derivatives are p-th power summable. Indiana Univ. Math. J. 84 (1972), 139–158.Google Scholar
  22. [F1]
    Fichera, G., Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Mem. Sez. I (8) 7 (1964), 71–140.Google Scholar
  23. [F2]
    Fichera, G., Boundary value problems in elasticity with unilateral constraints. Handbuch der Physik, VIa/2, Springer-Verlag, Berlin Heidelberg New York (1972), 347–389.Google Scholar
  24. [GT]
    Gastaldi, F., & Tomarelli, F., Some remarks on non-linear and non-coercive variational inequalities. Boll. Un. Mat. Ital. (7) 1-B (1987), 143–165.Google Scholar
  25. [GG]
    Giaquinta, M., & Giusti, E., Researches on the equilibrium of masonry structures. Arch. Rational Mech. Anal. 88 (1985), 359–392.Google Scholar
  26. [K1]
    Kinderlehrer, D., Remarks about Signorini's problem in linear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 8 (1981), 605–645.Google Scholar
  27. [K2]
    Kinderlehrer, D., Estimates for the solution and its stability in Signorini's problem. Appl. Math. Optim. 8 (1982), 159–188.Google Scholar
  28. [KS]
    Kinderlehrer, D., & Stampacchia, G., An introduction to variational inequalities and their applications. Academic Press, New York (1980).Google Scholar
  29. [LeS]
    Lewy, H., & Stampacchia, G., On the regularity of the solution of a variational inequality. Comm. Pure Appl. Math. 22 (1969), 153–188.Google Scholar
  30. [LM]
    Lions, J. L., & Magenes, E., Non-homogeneous boundary value problems and applications. Vol. 1, Springer-Verlag, Berlin Heidelberg New York (1972).Google Scholar
  31. [LS]
    Lions, L. J., & Stampacchia, G., Variational inequalities. Comm. Pure Appl. Math. 20 (1967), 493–519.Google Scholar
  32. [M1]
    Morrey, C. B. Jr., Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.Google Scholar
  33. [M2]
    Morrey, C. B. Jr., Multiple integrals in the calculus of variations. Springer-Verlag, Berlin Heidelberg New York (1966).Google Scholar
  34. [R]
    Rockafellar, R. T., Convex analysis. Princeton Univ. Press, Princeton (1970).Google Scholar
  35. [Sc]
    Schatzman, M., Problèmes aux limites non linéaires, non coercifs. Ann. Sc. Norm. Sup. Pisa Cl. Sci. III 27 (1973), 641–686.Google Scholar
  36. [Si]
    Signorini, A., Questioni di elasticità non linearizzata e semilinearizzata. Rend. Mat. 18 (1959), 95–139.Google Scholar
  37. [Te]
    Temam, R., Problèmes mathématiques en plasticité. Gauthier-Villars, Paris (1984).Google Scholar
  38. [TS]
    Temam, R., & Strang, G., Functions of bounded deformation. Arch. Rational Mech. Anal. 75 (1980), 7–21.Google Scholar
  39. [To1]
    Tonelli, L., Fondamenti di calcolo delle variazioni. Vols 1,2. Zanichelli, Bologna (1921, 1923).Google Scholar
  40. [To2]
    Tonelli, L., Opere scelte. Vols. 1, 2, 3, 4, Cremonese, Roma (1960, 1961, 1962, 1963).Google Scholar
  41. [V]
    Valent, T., Sulla formulazione variazionale — espressa nello stress — del problema dell'equilibrio dei corpi elastici con un vincolo di appoggio unilaterale liscio. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 55 (1974), 729–737.Google Scholar

Copyright information

© Springer-Verlag GmbH & Co 1988

Authors and Affiliations

  • Claudio Baiocchi
    • 1
    • 2
    • 3
  • Giuseppe Buttazzo
    • 1
    • 2
    • 3
  • Fabio Gastaldi
    • 1
    • 2
    • 3
  • Franco Tomarelli
    • 1
    • 2
    • 3
  1. 1.Dipartimento di Matematicadell'UniversitàPavia
  2. 2.Scuola Normale SuperiorePisa
  3. 3.Istituto di Analisi Numerica del C. N. R.Pavia

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