*Published Paper*

**Inserted:** 19 apr 2011

**Last Updated:** 4 sep 2012

**Journal:** J. Convex Anal.

**Volume:** 19

**Pages:** 225-248

**Year:** 2012

**Abstract:**

We consider the following classical autonomous variational problem \[\textrm{minimize} \left\{F(u)=\int_a^b f(u(x),u'(x))\ d x\,:\,u\in AC([a,b]), u(a)=\alpha, u(b)=\beta,\,u([a,b])\subseteq I \right\}\] where $I$ is a real interval, $\alpha, \beta\in I$, and $f:I\times \mathbb{R}\to [0,+\infty)$ is possibly neither continuous, nor coercive, nor convex; in particular $f(s,\cdot)$ may be not convex at $0$. Assuming the solvability of the relaxed problem, we prove under mild assumptions that the above variational problem has a solution, too.

**Keywords:**
non-convex variational problem, non-coercive variational problem, autonomous variational problem, relaxation result

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