Abstract
The problem of minimizing a functional on a subset of a Hilbert space is considered in a non-coercive and non-convex framework. Sufficient conditions for the existence of minima are given, involving a suitable recession functional. Some particular functionals are detailed, notably the quadratic ones, for which the existence theorem is specialized. Applications to the bending of a partially supported plate and to the classical Signorini problem are given. A modified interpretation of the unilateral condition for the Signorini problem is also introduced, suitable in finite elasticity. The abstract existence theorem is then applied to this concrete problem, providing sufficient conditions for the existence of an equilibrium configuration for an elastic body constrained to lie inside a box with rigid contour.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Signorini, A. Questioni di elasticità non linearizzata e semi-linearizzata. Rend. Mat., 18 (1959), 95–139.
Fichera G. Problemi elastostatici con vincoli unilaterali: il probiema di Signorini con ambigue condizioni al contorno. Atti Acc. Naz. Lin-cei Mem. Sez. I, (8) 7 (1964), 71–140.
Fichera, G. Boundary value problems in elasticity with unilateral constraints, Handbuch der Physik, Band VIa/2, Springer Verlag, Berlin (1972), 347–389.
Lions, J.L. and Stampacchia, G. Variational inequalities. Comm. Pure Appl. Math., 20 (1967), 493–519.
Schatzman, M. Problèmes aux limites non linéaires, non coercifs. Ann. Sc. Norm. Sup. Pisa, (3) 27 (1973), 641–686.
Baiocchi, C., Gastaldi, F. and Tomarelli, F. Inéquations variation-nelles non coercives. C.R. Ac. Sci. Paris., 299 (1984), 647–650.
Baiocchi, C., Gastaldi, F. and Tomarelli, F. Some existence results on non-coercive variational inequalities. Ann. Scuola Norm. Sup. Pisa (to appear).
Gastaldi, F. and Tomarelli, F. Some remarks on non-linear and noncoercive variational inequalities. Boll. Un. Mat. Ital. (to appear).
Kinderlehrer, D. Remarks about Signorini’s problem in linear elasticity. Ann. Scuola Norm. Sup. Pisa, 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.
Ball, J. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 63 (1977), 337–406.
Ciariet, P.G. and Necas, J. Unilateral problems in nonlinear, three dimensional elasticity. Arch. Rational Mech. Anal., 87 (1985), 319–338.
Ciariet, P.G. and Necas, J. Almost everywhere injectivity, self-contact and the non-vnterpenetration in non-linear, three-dimensional elasticity. (To appear).
Baiocchi, C., Buttazzo, G., Gastaldi, F. and Tomarelli, F. General existence results for unilateral problems in Continuum Mechanics. (To appear).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag Wien
About this paper
Cite this paper
Gastaldi, F. (1987). Existence Results for Minima of Non-Coercive Funtionals and Applications to Unilateral Problems in Elasticity. In: Del Piero, G., Maceri, F. (eds) Unilateral Problems in Structural Analysis — 2. International Centre for Mechanical Sciences, vol 304. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2967-8_9
Download citation
DOI: https://doi.org/10.1007/978-3-7091-2967-8_9
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-82036-0
Online ISBN: 978-3-7091-2967-8
eBook Packages: Springer Book Archive