2010
DOI: 10.1051/cocv/2010001
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Monotonicity properties of minimizers and relaxation for autonomous variational problems

Abstract: Abstract.We consider the following classical autonomous variational problemwhere the Lagrangian f is possibly neither continuous, nor convex, nor coercive. We prove a monotonicity property of the minimizers stating that they satisfy the maximum principle or the minimum one. By virtue of such a property, applying recent results concerning constrained variational problems, we derive a relaxation theorem, the DuBois-Reymond necessary condition and some existence or non-existence criteria.

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Cited by 4 publications
(8 citation statements)
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“…As for the necessary condition, we have proved in [9], subsequently improving the results in [10], that if (P ) is solvable then there exists a minimizer u satisfying the monotonicity property described above, such that ∂f (u(x), u (x)) = ∅ for a.e. x ∈ (a, b) and the DuBois-Reymond type necessary condition f u(x), u (x) − c ∈ u (x)∂f u(x), u (x) a.e.…”
Section: Introductionmentioning
confidence: 83%
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“…As for the necessary condition, we have proved in [9], subsequently improving the results in [10], that if (P ) is solvable then there exists a minimizer u satisfying the monotonicity property described above, such that ∂f (u(x), u (x)) = ∅ for a.e. x ∈ (a, b) and the DuBois-Reymond type necessary condition f u(x), u (x) − c ∈ u (x)∂f u(x), u (x) a.e.…”
Section: Introductionmentioning
confidence: 83%
“…Moreover, after a relaxation result, a new necessary and sufficient condition for the existence of the minimum of F is introduced, which is expressed in terms of an upper bound for the assigned slope β−α b−a . On the other hand, a certain monotonicity property of the minimizers of free problems has been recently studied in [9] and [10], where it was proved that, under very mild assumptions, the competition set Υ can be restricted, without loss of generality, to those trajectories admitting at most one change of monotonicity. More precisely, if (P ) is solvable, then it admits a minimizer which is increasing in [a, x 0 ] and decreasing in [x 0 , b] (or vice versa) for some x 0 ∈ [a, b].…”
Section: Introductionmentioning
confidence: 99%
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“…The definition and some properties of the cycloid are recalled in the Appendix. Further results on the existence of minimizers of non-coercive functionals are found, for instance, in [6,7,9,10].…”
Section: Introductionmentioning
confidence: 88%