Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 19 (2012), No. 1, 225--248Copyright Heldermann Verlag 2012 A Relaxation Result for Non-Convex and Non-Coercive Simple Integrals Massimiliano Bianchini Dip. di Matematica "U. Dini", Università di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy massimiliano.bianchini@math.unifi.it Giovanni Cupini Dip. di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy giovanni.cupini@unibo.it [Abstract-pdf] 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. Keywords: Non-convex variational problem, non-coercive variational problem, autonomous variational problem, relaxation result. MSC: 49K05,49J05 [ Fulltext-pdf  (198  KB)] for subscribers only.