Monotonicity properties of minimizers and relaxation for autonomous variational problems

Cupini, Giovanni; Marcelli, Cristina

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 222-242.

We consider the following classical autonomous variational problem

$$\mathrm{minimize}\left\{F\left(v\right)={\int }_a^{b}f(v\left(x\right),v^{\text{'}}\left(x\right))x̣:v\in AC\left([a,b]\right),v\left(a\right)=\alpha ,v\left(b\right)=\beta \right\},$$

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.

Classification : 49K05, 49J05
Mots clés : nonconvex variational problems, autonomous variational problems, existence of minimizers, Dubois-Reymond necessary condition, relaxation

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