Existence of Minimizers for Some Non Convex One-Dimensional Integrals

Nicola Fusco, Paolo Marcellini and António Ornelas

Abstract: We consider integrals of the type $\int_{a}^{b}\{h(u')+g(u)\}\,dx$, where $h$ is a nonconvex function such that $h^{**}(0)=h(0)$. It is still not known whether this condition alone on $h$ is sufficient to get existence of minimizers for general $g$. In this paper we prove it under very mild assumptions on $g$, e.g. it can be any combination of elementary functions.

Keywords: Calculus of variations; nonconvex integrals.

Classification (MSC2000): 49J05, 49K05, 49M20

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