*Published Paper*

**Inserted:** 8 jan 2010

**Last Updated:** 25 feb 2011

**Journal:** ESAIM Control Optim. Calc. Var.

**Volume:** 17

**Pages:** 222-242

**Year:** 2011

**Abstract:**

We consider the following classical autonomous variational problem \[\textrm{minimize\,} \left\{F(v)=\int_a^b f(v(x),v'(x))\ dx\,:\,v\in AC([a,b]), v(a)=\alpha, v(b)=\beta \right\},\] 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.

**Keywords:**
existence of minimizers, nonconvex variational problems, autonomous variational problems, DuBois-Reymond necessary condition

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