Monotonicity properties of minimizers and relaxation for autonomous variational problems
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 1, p. 222-242

We consider the following classical autonomous variational problem minimize F(v)= a b f(v(x),v ' (x))x̣:vAC([a,b]),v(a)=α,v(b)=β, where 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.

DOI : https://doi.org/10.1051/cocv/2010001
Classification:  49K05,  49J05
Keywords: nonconvex variational problems, autonomous variational problems, existence of minimizers, Dubois-Reymond necessary condition, relaxation
@article{COCV_2011__17_1_222_0,
     author = {Cupini, Giovanni and Marcelli, Cristina},
     title = {Monotonicity properties of minimizers and relaxation for autonomous variational problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     pages = {222-242},
     doi = {10.1051/cocv/2010001},
     zbl = {1213.49028},
     mrnumber = {2775194},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_1_222_0}
}
Cupini, Giovanni; Marcelli, Cristina. Monotonicity properties of minimizers and relaxation for autonomous variational problems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 1, pp. 222-242. doi : 10.1051/cocv/2010001. http://www.numdam.org/item/COCV_2011__17_1_222_0/

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