*Published Paper*

**Inserted:** 8 jan 2010

**Journal:** Ann. Inst. H. PoincarĂ© Anal. Non LinĂ©aire

**Volume:** 26

**Pages:** 1183-1205

**Year:** 2009

**Abstract:**

We consider the following autonomous variational problem \[ \mbox{minimize } \ \left\{ \int_a^b f(v(x),v'(x)) \ dx \ : \ v\in W^{1,1}(a,b),\ v(a)=\alpha, \ v(b)=\beta\right\}\] where the Lagrangian $f$ is assumed to be continuous, but not necessarily coercive, nor convex. We show that the existence of the minimum is linked to the solvability of certain constrained variational problems. This allows us to derive existence theorems covering a wide class of nonconvex noncoercive problems.

**Keywords:**
relaxation, Nonconvex problems, noncoercive problems, autonomous Lagrangians

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