Existence of Scalar Minimizers for Nonconvex Simple Integrals of Sum Type

Necessary and sufficient conditions for optimality of nonconvex, noncoercive autonomous variational problems with constraints

“…Under the same assumptions of Theorem 12, if c 0 < µ and In view of what was just observed, it is interesting to establish conditions ensuring the validity of condition (20), in such a way that the minimum exists for every positive assigned slope ξ 0 = β−α b−a . The next result provides an answer to this question.…”

“…In particular, in [11] and [20], Fusco, Marcellini and Ornelas proved the solvability of nonconvex but coercive autonomous integrals of sum type, under the assumption that the convex envelope coincides with the integrand at the origin. We quote also the recent result [5] for nonconvex autonomous multiple integrals.…”

Monotonicity properties of minimizers and relaxation for autonomous variational problems

“…It improves the analogous ones proved in [17,22] To present the proof of Theorem 4.2 we need the following auxiliary result, which states that under certain conditions it is possible to modify the minimizer of the relaxed problem in such a way that its derivative is far from 0 in the intervals where the trajectory is strictly monotone. …”

Nonconvex Minimization Problems for Functionals Defined on Vector Valued Functions

“…The existence of solutions to 1 in the scalar one-dimensional coercive w x case, i.e., n s m s 1 and M s qϱ, was recently studied in 12, 15 . In particular, it was proved that, if h is a lower semicontinuous function, Ž .…”