Geodesic
A Relaxation Result for Non-Convex and Non-Coercive Simple Integrals
Journal of convex analysis, Tome 19 (2012) no. 1, pp. 225-248

Voir la notice de l'article provenant de la source Heldermann Verlag

We consider the following classical autonomous variational problem: Minimize \[\left\{F(u)=\int_a^b f(u(x),u'(x))\,dx\,:\,u\in AC([a,b]), u(a)=\alpha, u(b)=\beta,\,u([a,b]) \subseteq I \right\}\] 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.
Classification : 49K05,49J05
Mots-clés : Non-convex variational problem, non-coercive variational problem, autonomous variational problem, relaxation result 
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     author = {M. Bianchini and G. Cupini},
     title = {A {Relaxation} {Result} for {Non-Convex} and {Non-Coercive} {Simple} {Integrals}},
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     year = {2012},
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M. Bianchini; G. Cupini. A Relaxation Result for Non-Convex and Non-Coercive Simple Integrals. Journal of convex analysis, Tome 19 (2012) no. 1, pp. 225-248. http://geodesic.mathdoc.fr/item/JCA_2012_19_1_JCA_2012_19_1_a13/