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Journal of Convex Analysis 19 (2012), No. 1, 225--248
Copyright Heldermann Verlag 2012

A Relaxation Result for Non-Convex and Non-Coercive Simple Integrals

Massimiliano Bianchini
Dip. di Matematica "U. Dini", UniversitÓ di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy

Giovanni Cupini
Dip. di Matematica, UniversitÓ di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy


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

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