Calculus of Variations and Geometric Measure Theory

M. Bianchini - G. Cupini

A relaxation result for non-convex and non-coercive simple integrals

created by cupini on 19 Apr 2011
modified on 04 Sep 2012


Published Paper

Inserted: 19 apr 2011
Last Updated: 4 sep 2012

Journal: J. Convex Anal.
Volume: 19
Pages: 225-248
Year: 2012


We consider the following classical autonomous variational problem \[\textrm{minimize} \left\{F(u)=\int_a^b f(u(x),u'(x))\ d x\,:\,u\in AC([a,b]), u(a)=\alpha, u(b)=\beta,\,u([a,b])\subseteq I \right\}\] 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