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.
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References
Acerbi, E., & Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984), 125–145.
Anzellotti, G., A class of convex non-coercive functionals and masonry-like materials. Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 261–307.
Anzellotti, G., Buttazzo, G., & Dal Maso, G., Dirichlet problems for demicoercive functionals. Nonlinear Anal. 10 (1986), 603–613.
Baiocchi, C., Disequazioni variazionali non coercive. Proc. Convegno Intern. M. Picone-L. Tonelli, Roma (1985) (to appear).
Baiocchi, C., & Capelo, A., Variational and quasivariational inequalities: applications to free boundary problems. J. Wiley and Sons, Chichester (1984).
Baiocchi, C., Gastaldi, F., & Tomarelli, F., Inéquations variationnelles non coercives. C. R. Acad. Sci. Paris. Ser. I Math. 299 (1984), 647–650.
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.
Ball, J., Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–406.
Boieri, P., Gastaldi, F., & Kinderlehrer, D., Existence, uniqueness and regularity results for the two bodies contact problem. Appl. Math. Optim. 15 (1987), 251–277.
Bourbaki, N., Eléments de Mathématique—Espaces Vectoriels Topologiques, Ch. 1 et 2. Act. Sci. Ind., 1189, Hermann, Paris (1966).
Busemann, H., Ewald, G., & Shepard, G. C., Convex bodies and convexity on Grassmann cones. Parts I–IV, Math. Ann. 151 (1963), 1–41.
Buttazzo, G., Su una definizione generate dei Γ-limiti. Boll. Un. Mat. Ital. (5) 14-B (1977), 722–744.
Ciarlet, P. G., Elasticité tridimensionnelle. Masson, Paris (1986).
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.
Ciarlet, P. G., & Nečas, J., Unilateral problems in nonlinear, three dimensional elasticity. Arch. Rational Mech. Anal. 87 (1985), 319–338.
Ciarlet, P. G., & Nečas, J., Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987), 171–188.
Dal Maso, G., & Longo, P., Γ-limits of obstacles. Ann. Mat. Pura Appl. 128 (1980), 1–50.
De Giorgi, E., Γ-convergenza e G-convergenza. Boll. Un. Mat. Ital. (5) 14-A (1977), 213–224.
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.
Duvaut, G., & Lions, J. L., Inequalities in mechanics and physics. Springer-Verlag, Berlin Heidelberg New York (1976).
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.
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.
Fichera, G., Boundary value problems in elasticity with unilateral constraints. Handbuch der Physik, VIa/2, Springer-Verlag, Berlin Heidelberg New York (1972), 347–389.
Gastaldi, F., & Tomarelli, F., Some remarks on non-linear and non-coercive variational inequalities. Boll. Un. Mat. Ital. (7) 1-B (1987), 143–165.
Giaquinta, M., & Giusti, E., Researches on the equilibrium of masonry structures. Arch. Rational Mech. Anal. 88 (1985), 359–392.
Kinderlehrer, D., Remarks about Signorini's problem in linear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 8 (1981), 605–645.
Kinderlehrer, D., Estimates for the solution and its stability in Signorini's problem. Appl. Math. Optim. 8 (1982), 159–188.
Kinderlehrer, D., & Stampacchia, G., An introduction to variational inequalities and their applications. Academic Press, New York (1980).
Lewy, H., & Stampacchia, G., On the regularity of the solution of a variational inequality. Comm. Pure Appl. Math. 22 (1969), 153–188.
Lions, J. L., & Magenes, E., Non-homogeneous boundary value problems and applications. Vol. 1, Springer-Verlag, Berlin Heidelberg New York (1972).
Lions, L. J., & Stampacchia, G., Variational inequalities. Comm. Pure Appl. Math. 20 (1967), 493–519.
Morrey, C. B. Jr., Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.
Morrey, C. B. Jr., Multiple integrals in the calculus of variations. Springer-Verlag, Berlin Heidelberg New York (1966).
Rockafellar, R. T., Convex analysis. Princeton Univ. Press, Princeton (1970).
Schatzman, M., Problèmes aux limites non linéaires, non coercifs. Ann. Sc. Norm. Sup. Pisa Cl. Sci. III 27 (1973), 641–686.
Signorini, A., Questioni di elasticità non linearizzata e semilinearizzata. Rend. Mat. 18 (1959), 95–139.
Temam, R., Problèmes mathématiques en plasticité. Gauthier-Villars, Paris (1984).
Temam, R., & Strang, G., Functions of bounded deformation. Arch. Rational Mech. Anal. 75 (1980), 7–21.
Tonelli, L., Fondamenti di calcolo delle variazioni. Vols 1,2. Zanichelli, Bologna (1921, 1923).
Tonelli, L., Opere scelte. Vols. 1, 2, 3, 4, Cremonese, Roma (1960, 1961, 1962, 1963).
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.
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Baiocchi, C., Buttazzo, G., Gastaldi, F. et al. General existence theorems for unilateral problems in continuum mechanics. Arch. Rational Mech. Anal. 100, 149–189 (1988). https://doi.org/10.1007/BF00282202
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DOI: https://doi.org/10.1007/BF00282202