On mimima of a functional of the gradient: Sufficient conditions. (English) Zbl 0784.49022

This paper is a sequel to a previous paper of the author [ibid. 20, No. 4, 337-341 (1993; Zbl 0784.49021)]. It shows that the geometrical condition, which was shown to be necessary in the first paper, is also sufficient for the existence of a solution. Thus, the two papers yield necessary and sufficient conditions for the existence of a solution for a nonconvex integrand and necessary and sufficient conditions for the uniqueness of the solution for the case of a convex integrand.


49K99 Optimality conditions


Zbl 0784.49021
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[1] Cellina, A., On minima of a functional of the gradient: necessary conditions, Nonlinear Analysis, 20, 4, 337-341 (1993) · Zbl 0784.49021
[2] Marcellini, P., A relation between existence of minima for non convex integrals and uniqueness for non strictly convex integrals of the Calculus of Variations, (Lecture Notes in Mathematics, Vol. 979 (1983), Springer: Springer Berlin), 216-232
[3] Rockafellar, R. T., Convex Analysis (1972), Princeton University Press: Princeton University Press Princeton, New Jersey · Zbl 0224.49003
[4] Ekeland, I.; Temam, R., Convex Analysis and Variational Problems (1976), North Holland: North Holland Amsterdam
[5] Gurtin, M. E., An Introduction to Continuum Mechanics (1981), Academic Press: Academic Press New York · Zbl 0559.73001
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