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.

We consider the following classical autonomous variational problem

 $\mathrm{minimize}\phantom{\rule{0.166667em}{0ex}}\left\{F\left(v\right)={\int }_{a}^{b}f\left(v\left(x\right),{v}^{\text{'}}\left(x\right)\right)\phantom{\rule{4pt}{0ex}}x̣\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}v\in AC\left(\left[a,b\right]\right),\phantom{\rule{0.277778em}{0ex}}v\left(a\right)=\alpha ,\phantom{\rule{0.277778em}{0ex}}v\left(b\right)=\beta \right\},$
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
Keywords: 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, Volume 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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