Monotonicity properties of minimizers and relaxation for autonomous variational problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 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 : 10.1051/cocv/2010001
Classification : 49K05, 49J05
Mots clés : nonconvex variational problems, autonomous variational problems, existence of minimizers, Dubois-Reymond necessary condition, relaxation
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Cupini, Giovanni; Marcelli, Cristina. Monotonicity properties of minimizers and relaxation for autonomous variational problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 222-242. doi : 10.1051/cocv/2010001. http://www.numdam.org/articles/10.1051/cocv/2010001/

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