On the Minimum Problem for a Class of Non-coercive Functionals

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

“…Under this assumption they proved some conditions guaranteeing the solvability of problem (P ) related to the solvability of (P * * ) (see also [7] and [10] for similar results). In [6], Cellina considered a pointwise version of condition (1) (see assumption (GA)), by avoiding the requirement of uniformity, and proved relaxation results and the Lipschitz continuity of minimizers in the vectorial case.…”

“…Then, similarly to what was done in the proof of property i), we can find a decreasing sequence (t n ) n , converging to w v (τ 0 ), such that t n = w v (v(t n )), n ∈ N. Therefore, inf in (α, β). By the absolute continuity of v, we have that |v(A v )| = β − α, where A v was defined in (7). So, for a.e.…”

Minimizing Nonconvex, Simple Integrals of Product Type

“…Finally, we recall a very weak growth condition on f ** (see [1] and [3]) that will play a preminent ro^le in the following. To this purpose, recall that whenever d # R is a subgradient of f ** at some point !…”

Some Problems in the Calculus of Variations

“…Assumption 1, in a slightly different form, was introduced in [10] for a problem of slow growth on the parametrizations of a given curve; in [3], it was shown that Assumption 1 implies that solutions to an autonomous functional defined over a one-dimensional set are Lipschitzian; in [7] it is shown that under Assumption 1, an autonomous functional as before actually admits solutions, through a variant of the Direct method; in fact, no coercivity is assumed; it is proved that, when Assumption 1 is satisfied, from a minimizing sequence we can select a subsequence such that suitable reparametrizations of the subsequence converge and still form a minimizing sequence. Finally, the problem on a multi-dimensional space was met in [9].…”